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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 2

Vector Spaces and Bases - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that the set of complex numbers $x+\mathrm{i} y$ forms a real vector space under the operations of addition $(x+\mathrm{i} y)+(u+\mathrm{i} v)=(x+u)+\mathrm{i}(y+v)$ and scalar multiplication $c(x+\mathrm{i} y)=c x+\mathrm{i} c y$. (But complex multiplication is not a real vector space operation.)

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08:32

Problem 1

(a) Prove that the set of all vectors $(x, y, z)^T$ such that $x-y+4 z=0$ forms a subspace of $\mathbb{R}^3$. (b) Explain why the set of all vectors that satisfy $x-y+4 z=1$ does not form a subspace.

Donald Albin
Donald Albin
Numerade Educator
08:32

Problem 1

Show that $\left(\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right)$ belongs to the subspace of $\mathbb{R}^3$ spanned by $\left(\begin{array}{r}2 \\ -1 \\ 2\end{array}\right),\left(\begin{array}{r}5 \\ -4 \\ 1\end{array}\right)$ by writing it as a linear combination of the spanning vectors.

Donald Albin
Donald Albin
Numerade Educator

Problem 1

Determine which of the following sets of vectors are bases of $\mathbb{R}^2$ :
(a) $\left(\begin{array}{r}1 \\ -3\end{array}\right),\left(\begin{array}{r}-2 \\ 5\end{array}\right)$
(b) $\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$;
(c) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 1\end{array}\right)$;
(d) $\left(\begin{array}{l}3 \\ 5\end{array}\right),\left(\begin{array}{l}0 \\ 0\end{array}\right)$;
(e) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

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01:50

Problem 1

Characterize the image and kernel of the following matrices:
(a)
$\left(\begin{array}{rr}8 & -4 \\ -6 & 3\end{array}\right)$
(b)
$\left(\begin{array}{rrr}1 & -1 & 2 \\ -2 & 2 & -4\end{array}\right)$
(c) $\left(\begin{array}{rrr}1 & 2 & 3 \\ -2 & 4 & 1 \\ 4 & 0 & 5\end{array}\right)$,
(d)
$$
\left(\begin{array}{rrrr}
1 & -1 & 0 & 1 \\
-1 & 0 & 1 & -1 \\
1 & -2 & 1 & 1 \\
1 & 2 & -3 & 1
\end{array}\right)
$$

Thomas Emment
Thomas Emment
Numerade Educator
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Problem 1

(a) Draw the graph corresponding to the $6 \times 7$ incidence matrix whose nonzero $(i, j)$ entries equal 1 if $j=i$ and -1 if $j=i+1$, for $i=1$ to 6 . (b) Find a basis for its kernel and cokernel. (c) How many circuits are in the digraph?

Nick Johnson
Nick Johnson
Numerade Educator
02:50

Problem 2

Show that the positive quadrant $Q=\{(x, y) \mid x, y>0\} \subset \mathbb{R}^2$ forms a vector space if we define addition by $\left(x_1, y_1\right)+\left(x_2, y_2\right)=\left(x_1 x_2, y_1 y_2\right)$ and scalar multiplication by $c(x, y)=\left(x^c, y^c\right)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
10:24

Problem 2

Which of the following are subspaces of $\mathbb{R}^3$ ? Justify your answers! (a) The set of all vectors $(x, y, z)^T$ satisfying $x+y+z+1=0$. (b) The set of vectors of the form $(t,-t, 0)^T$ for $t \in \mathbb{R}$. (c) The set of vectors of the form $(r-s, r+2 s,-s)^T$ for $r, s \in \mathbb{R}$. (d) The set of vectors whose first component equals 0 . (e) The set of vectors whose last component equals 1. (f) The set of all vectors $(x, y, z)^T$ with $x \geq y \geq z$. (g) The set of all solutions to the equation $z=x-y$. (h) The set of all solutions to $z=x y$. (i) The set of all solutions to $x^2+y^2+z^2=0$. (j) The set of all solutions to the system $x y=y z=x z$.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
02:51

Problem 2

Show that $\left(\begin{array}{r}-3 \\ 7 \\ 6 \\ 1\end{array}\right)$ is in the subspace of $\mathbb{R}^4$ spanned by $\left(\begin{array}{r}1 \\ -3 \\ -2 \\ 0\end{array}\right),\left(\begin{array}{r}-2 \\ 6 \\ 3 \\ 4\end{array}\right)$ and $\left(\begin{array}{r}-2 \\ 4 \\ 6 \\ -7\end{array}\right)$.

Daniel Pezzi
Daniel Pezzi
Numerade Educator
05:43

Problem 2

Determine which of the following are bases of $\mathbb{R}^3$ :
(a) $\left(\begin{array}{l}2 \\ 1 \\ 5\end{array}\right),\left(\begin{array}{l}1 \\ 5 \\ 2\end{array}\right)$
(b) $\left(\begin{array}{r}0 \\ 1 \\ -5\end{array}\right)$,
$$
\left(\begin{array}{r}
-1 \\
3 \\
0
\end{array}\right),\left(\begin{array}{l}
1 \\
3 \\
0
\end{array}\right)
$$
(c)
$$
\left(\begin{array}{r}
0 \\
4 \\
-1
\end{array}\right),\left(\begin{array}{r}
-1 \\
0 \\
1
\end{array}\right),\left(\begin{array}{r}
1 \\
-8 \\
1
\end{array}\right)
$$
$$
\left(\begin{array}{r}
2 \\
0 \\
-2
\end{array}\right),\left(\begin{array}{r}
-1 \\
2 \\
-1
\end{array}\right),\left(\begin{array}{r}
0 \\
-1 \\
0
\end{array}\right),\left(\begin{array}{r}
-1 \\
2 \\
1
\end{array}\right) \text {. }
$$

Victor Salazar
Victor Salazar
Numerade Educator

Problem 2

For the following matrices, write the kernel as the span of a finite number of vectors. Is the kernel a point, line, plane, or all of $\mathbb{R}^3$ ?
(a) $\left(\begin{array}{ll}2 & -1\end{array}\right.$ 5),
(b) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 3 & -2 & 0\end{array}\right)$,
(c)
$\left(\begin{array}{rrr}2 & 6 & -4 \\ -1 & -3 & 2\end{array}\right)$
(d)
$\left(\begin{array}{rrr}1 & 2 & 5 \\ 0 & 4 & 8 \\ 1 & -6 & -11\end{array}\right)$
(e) $\left(\begin{array}{rrr}2 & -1 & 1 \\ -1 & 1 & -2 \\ 3 & -1 & 1\end{array}\right)$,
(f)
$\left(\begin{array}{rrr}1 & -2 & 3 \\ -3 & 6 & -9 \\ -2 & 4 & -6 \\ 3 & 0 & -1\end{array}\right)$

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11:05

Problem 2

Draw the digraph represented by the following incidence matrices:
(a) $\left(\begin{array}{rrrr}-1 & 0 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & -1 & 1 & 0 \\ 0 & 1 & 0 & -1\end{array}\right)$,
(b) $\left(\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right)$,
(c) $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & -1 & 1 & 0 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrrrr}-1 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrrrrrr}0 & 1 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1\end{array}\right)$.

Chris Trentman
Chris Trentman
Numerade Educator
15:42

Problem 3

Let $S$ be any set. Carefully justify the validity of all the vector space axioms for the space $\mathcal{F}(S)$ consisting of all real-valued functions $f: S \rightarrow \mathbb{R}$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:04

Problem 3

Graph the following subsets of $\mathbb{R}^3$ and use this to explain which are subspaces:
(a) The line $(t,-t, 3 t)^T$ for $t \in \mathbb{R}$. (b) The helix $(\cos t, \sin t, t)^T$. (c) The surface $x-2 y+3 z=0$. (d) The unit ball $x^2+y^2+z^2<1$. (e) The cylinder $(y+2)^2+(z-1)^2=5$.
(f) The intersection of the cylinders $(x-1)^2+y^2=1$ and $(x+1)^2+y^2=1$.

Carson Merrill
Carson Merrill
Numerade Educator
03:30

Problem 3

(a) Determine whether $\left(\begin{array}{r}1 \\ -2 \\ -3\end{array}\right)$ is in the span of $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ and $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right) . \quad$ (b) Is $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3 \\ 4\end{array}\right)$ ?
(c) Is $\left(\begin{array}{r}3 \\ 0 \\ -1 \\ -2\end{array}\right)$ in the span of $\left(\begin{array}{l}1 \\ 2 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 3 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ 0 \\ 1 \\ -1\end{array}\right)$ ?

Donald Albin
Donald Albin
Numerade Educator
04:48

Problem 3

Let $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 0 \\ 2\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}3 \\ -1 \\ 1\end{array}\right), \mathbf{v}_3=\left(\begin{array}{r}2 \\ -1 \\ -1\end{array}\right), \mathbf{v}_4=\left(\begin{array}{r}4 \\ -1 \\ 3\end{array}\right)$.
(a) Do $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ span $\mathbb{R}^3$ ? Why or why not?
(b) Are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ linearly independent? Why or why not? (c) Do $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ form a basis for $\mathbb{R}^3$ ? Why or why not? If not, is it possible to choose some subset that is a basis?
(d) What is the dimension of the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ ? Justify your answer.

ET
Ed Tam
Numerade Educator
01:52

Problem 3

(a) Find the kernel and image of the coefficient matrix for the system $x-3 y+2 z=a$, $2 x-6 y+2 w=b, z-3 w=c$. (b) Write down compatibility conditions on $a, b, c$ for a solution to exist.

Mark Augustyn
Mark Augustyn
Numerade Educator
00:30

Problem 3

Write out the incidence matrix of the following digraphs.
(a)
(b)
(c)
(d)
(e)
(f)

Kyle Christian
Kyle Christian
Numerade Educator
05:24

Problem 4

Let $S=\{0,1,2,3\}$. (a) Find the sample vectors corresponding to the functions 1 , $\cos \pi x, \cos 2 \pi x, \cos 3 \pi x$. (b) Is a function uniquely determined by its sample values?

Harshita Goel
Harshita Goel
Numerade Educator
08:32

Problem 4

Show that if $W \subset \mathbb{R}^3$ is a subspace containing the vectors $(1,2,-1)^T,(2,0,1)^T$, $(0,-1,3)^T$, then $W=\mathbb{R}^3$.

Donald Albin
Donald Albin
Numerade Educator
08:11

Problem 4

Which of the following sets of vectors span all of $\mathbb{R}^2$ ? (a) $\left(\begin{array}{r}1 \\ -1\end{array}\right) ;(b)\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{l}1 \\ 3\end{array}\right)$;
(c) $\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 2\end{array}\right)$;
(d) $\left(\begin{array}{r}6 \\ -9\end{array}\right),\left(\begin{array}{r}-4 \\ 6\end{array}\right)$;
(e) $\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{r}3 \\ -1\end{array}\right)$;
; $(f)\left(\begin{array}{l}0 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ -1\end{array}\right),\left(\begin{array}{r}2 \\ -2\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
03:40

Problem 4

Answer Exercise 2.4.3 when $\mathbf{v}_1=\left(\begin{array}{r}1 \\ -1 \\ 2\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}2 \\ -2 \\ 5\end{array}\right), \mathbf{v}_3=\left(\begin{array}{r}0 \\ -2 \\ 1\end{array}\right), \mathbf{v}_4=\left(\begin{array}{r}1 \\ 3 \\ -1\end{array}\right)$.

James Chok
James Chok
Numerade Educator
04:41

Problem 4

Suppose $\mathbf{x}^{\star}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ is a particular solution to the equation $\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1\end{array}\right) \mathbf{x}=\mathbf{b}$.
(a) What is $\mathbf{b}$ ?
(b) Find the general solution.

Ryan Williams
Ryan Williams
Numerade Educator
21:05

Problem 4

For each of the digraphs in Exercise 2.6.3, see whether you can predict a collection of independent circuits. Verify your prediction by constructing a suitable basis of the cokernel of the incidence matrix and identifying each basis vector with a circuit.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
01:42

Problem 5

Find two different functions $f(x)$ and $g(x)$ that have the same sample vectors $\mathbf{f}, \mathbf{g}$ at the sample points $x_1=0, x_2=1, x_3=-1$.

Melissa Barry
Melissa Barry
Numerade Educator
03:38

Problem 5

True or false: An interval is a vector space.

Prashant Bana
Prashant Bana
Numerade Educator
01:29

Problem 5

(a) Graph the subspace of $\mathbb{R}^3$ spanned by the vector $\mathbf{v}_1=(3,0,1)^T$.
(b) Graph the subspace spanned by the vectors $\mathbf{v}_1=(3,-2,-1)^T, \mathbf{v}_2=(-2,0,-1)^T$.
(c) Graph the span of $\mathbf{v}_1=(1,0,-1)^T, \mathbf{v}_2=(0,-1,1)^T, \mathbf{v}_3=(1,-1,0)^T$.

Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Vvn1Eec8Hpzl08Ivucuckdn8Igliwh6 Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Yfcjxtp7V4Zxsmgv8Xpg.Vn.Fy6Khx6
Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Vvn1Eec8Hpzl08Ivucuckdn8Igliwh6 Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Yfcjxtp7V4Zxsmgv8Xpg.Vn.Fy6Khx6
Numerade Educator
02:03

Problem 5

Find a basis for (a) the plane given by the equation $z-2 y=0$ in $\mathbb{R}^3 ;(b)$ the plane given by the equation $4 x+3 y-z=0$ in $\mathbb{R}^3 ;(c)$ the hyperplane $x+2 y+z-w=0$ in $\mathbb{R}^4$.

Victor Salazar
Victor Salazar
Numerade Educator
01:49

Problem 5

Prove that the average of all the entries in each row of $A$ is 0 if and only if $(1,1, \ldots, 1)^T \in \operatorname{ker} A$.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 5

(a) Write down the incidence matrix $A$ for the indicated digraph.
(b) What is the rank of $A$ ? (c) Determine the dimensions of its four fundamental subspaces. (d) Find a basis for its kernel and cokernel.
(e) Determine explicit conditions on vectors $\mathbf{b}$ that guarantee that the system $A \mathbf{x}=\mathbf{b}$ has a solution. (f) Write down a specific nonzero vector $\mathbf{b}$ that satisfies your conditions, and then find all possible solutions.

Victor Salazar
Victor Salazar
Numerade Educator
04:38

Problem 6

(a) Let $x_1=0, x_2=1$. Find the unique linear function $f(x)=a x+b$ that has the sample vector $\mathbf{f}=(3,-1)^T$. (b) Let $x_1=0, x_2=1, x_3=-1$. Find the unique quadratic function $f(x)=a x^2+b x+c$ with sample vector $\mathbf{f}=(1,-2,0)^T$.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
11:42

Problem 6

(a) Can you construct an example of a subset $S \subset \mathbb{R}^2$ with the property that $c \mathbf{v} \in S$ for all $c \in \mathbb{R}, \mathbf{v} \in S$, and yet $S$ is not a subspace? (b) What about an example in which $\mathbf{v}+\mathbf{w} \in S$ for every $\mathbf{v}, \mathbf{w} \in S$, and yet $S$ is not a subspace?

E R
E R
Numerade Educator
01:47

Problem 6

Let $U$ be the subspace of $\mathbb{R}^3$ spanned by $\mathbf{u}_1=(1,2,3)^T, \mathbf{u}_2=(2,-1,0)^T$. Let $V$ be the subspace spanned by $\mathbf{v}_1=(5,0,3)^T, \mathbf{v}_2=(3,1,3)^T$. Is $V$ a subspace of $U$ ? Are $U$ and $V$ the same?

Donald Albin
Donald Albin
Numerade Educator
01:38

Problem 6

(a) Show that $\left(\begin{array}{l}4 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 0\end{array}\right)$, and $\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -1\end{array}\right)$ are two different bases for the plane $x-2 y-4 z=0$. (b) Show how to write both elements of the second basis as linear combinations of the first. (c) Can you find a third basis?

Victor Salazar
Victor Salazar
Numerade Educator
00:26

Problem 6

True or false: If $A$ is a square matrix, then $\operatorname{ker} A \cap \operatorname{img} A=\{0\}$.

Taylor Shimono
Taylor Shimono
Numerade Educator
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Problem 6

(a) Write out the incidence matrix for the cubical digraph and identify the basis of its cokernel with the circuits. (b) Find three circuits that do not correspond to any of your basis elements, and express them as a linear combination of the basis circuit vectors.

Victor Salazar
Victor Salazar
Numerade Educator
06:21

Problem 7

Let $\mathcal{F}\left(\mathbb{R}^2, \mathbb{R}^2\right)$ denote the vector space consisting of all functions $\mathbf{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.
(a) Which of the following functions $\mathbf{f}(x, y)$ are elements?
(iii) $\left(\begin{array}{c}e^x \\ \cos y\end{array}\right)$,
(iv) $\left(\begin{array}{l}1 \\ 3\end{array}\right)$
(v) $\left(\begin{array}{cc}x & y \\ -y & x\end{array}\right)$,
(vi) $\left(\begin{array}{c}x \\ y \\ x+y\end{array}\right)$.
(i) $x^2+y^2,($ ii $)\left(\begin{array}{c}x-y \\ x y\end{array}\right)$,
(b) Sum all of the elements
of $\mathcal{F}\left(\mathbb{R}^2, \mathbb{R}^2\right)$ you identified in part (a). Then multiply your sum by the scalar -5 .
(c) Carefully describe the zero element of the vector space $\mathcal{F}\left(\mathbb{R}^2, \mathbb{R}^2\right)$.

WM
William Mead
Numerade Educator
01:50

Problem 7

Determine which of the following sets of vectors $\mathbf{x}=\left(x_1, x_2, \ldots, x_n\right)^T$ are subspaces of $\mathbb{R}^n$ : (a) all equal entries $x_1=\cdots=x_n ;(b)$ all positive entries: $x_i \geq 0 ;(c)$ first and last entries equal to zero: $x_1=x_n=0 ;(d)$ entries add up to zero: $x_1+\cdots+x_n=0$; (e) first and last entries differ by one: $x_1-x_n=1$.

Nick Johnson
Nick Johnson
Numerade Educator
02:29

Problem 7

(a) Let $S$ be the subspace of $\mathcal{M}_{2 \times 2}$ consisting of all symmetric $2 \times 2$ matrices. Show that $S$ is spanned by the matrices $\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$, and $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$.
(b) Find a spanning set of the space of symmetric $3 \times 3$ matrices.

Harshita Goel
Harshita Goel
Numerade Educator
01:16

Problem 7

A basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ is called right-handed if the $n \times n$ matrix $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ whose columns are the basis vectors has positive determinant: $\operatorname{det} A>0$. If $\operatorname{det} A<0$, the basis is called left-handed. (a) Which of the following form right-handed bases of $\mathbb{R}^3$ ?
(i) $\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$,
(ii) $\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)$
(iii) $\left(\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ -2\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right)$ (iv) $\left(\begin{array}{l}3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)$.
(b) Show that if $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ is a left-handed basis of $\mathbb{R}^3$, then $\mathbf{v}_2$, $\mathbf{v}_1, \mathbf{v}_3$ and $-\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are both right-handed bases. (c) What sort of basis has $\operatorname{det} A=0$ ?

Victor Salazar
Victor Salazar
Numerade Educator
02:17

Problem 7

Write the general solution to the following linear systems in the form (2.27). Clearly identify the particular solution $\mathbf{x}^{\star}$ and the element $\mathbf{z}$ of the kernel. (a) $x-y+3 z=1$,
(b) $\left(\begin{array}{rrr}1 & -2 & 0 \\ 2 & 3 & 1\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}3 \\ -1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & -1 & 0 \\ 2 & 0 & -4 \\ 2 & -1 & -2\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}-1 \\ -6 \\ -4\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & -1 & 1 \\ 4 & -1 & 2 \\ 0 & 1 & 3\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{rr}1 & -2 \\ 2 & -4 \\ -3 & 6 \\ -1 & 2\end{array}\right)\left(\begin{array}{l}u \\ v\end{array}\right)=\left(\begin{array}{r}-1 \\ -2 \\ 3 \\ 1\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}1 & -3 & 2 & 0 \\ -1 & 5 & 1 & 1 \\ 2 & -8 & 1 & -1\end{array}\right)\left(\begin{array}{l}p \\ q \\ r \\ s\end{array}\right)=\left(\begin{array}{r}4 \\ -3 \\ 7\end{array}\right)$,
(g)
$$
\left(\begin{array}{rrrr}
0 & -1 & 2 & -1 \\
1 & -3 & 0 & 1 \\
-2 & 5 & 2 & -3 \\
1 & 1 & -8 & 5
\end{array}\right)\left(\begin{array}{c}
x \\
y \\
z \\
w
\end{array}\right)=\left(\begin{array}{r}
-2 \\
-3 \\
4 \\
5
\end{array}\right)
$$

Tony Ni
Tony Ni
Numerade Educator
00:27

Problem 7

Write out the incidence matrix for the other Platonic solids: (a) tetrahedron, (b) octahedron, (c) dodecahedron, and (d) icosahedron. (You will need to choose an orientation for the edges.) Show that, in each case, the number of independent circuits equals the number of faces minus 1.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:28

Problem 8

A planar vector field is a function that assigns a vector $\mathbf{v}(x, y)=\left(\begin{array}{l}v_1(x, y) \\ v_2(x, y)\end{array}\right)$ to each point $\left(\begin{array}{l}x \\ y\end{array}\right) \in \mathbb{R}^2$. Explain why the set of all planar vector fields forms a vector space.

Harshita Goel
Harshita Goel
Numerade Educator
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Problem 8

Prove that the set of all solutions $\mathbf{x}$ of the linear system $A \mathbf{x}=\mathbf{b}$ forms a subspace if and only if the system is homogeneous.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 8

(a) Determine whether the polynomials $x^2+1, x^2-1, x^2+x+1 \operatorname{span} \mathcal{P}^{(2)}$.
(b) Do $x^3-1, x^2+1, x-1,1$ span $\mathcal{P}^{(3)}$ ? (c) What about $x^3, x^2+1, x^2-x, x+1$ ?

Victor Salazar
Victor Salazar
Numerade Educator
05:49

Problem 8

Find a basis for and the dimension of the following subspaces: (a) The space of solutions to the linear system $A \mathbf{x}=\mathbf{0}$, where $A=\left(\begin{array}{rrrr}1 & 2 & -1 & 1 \\ 3 & 0 & 2 & -1\end{array}\right)$.
(b) The set of all quadratic polynomials $p(x)=a x^2+b x+c$ that satisfy $p(1)=0$. (c) The space of all solutions to the homogeneous ordinary differential equation $u^{\prime \prime \prime}-u^{\prime \prime}+4 u^{\prime}-4 u=0$.

Barsha Rana
Barsha Rana
Numerade Educator
02:28

Problem 8

Given $a, r \neq 0$, characterize the kernel and the image of the matrix
$$
\left(\begin{array}{cccc}
a & a r & \ldots & a r^{n-1} \\
a r^n & a r^{n+1} & \ldots & a r^{2 n-1} \\
\vdots & \vdots & \ddots & \vdots \\
a r^{(n-1) n} & a r^{(n-1) n+1} & \ldots & a r^{n^2-1}
\end{array}\right) \text { }
$$

AG
Ankit Gupta
Numerade Educator
04:04

Problem 8

Prove that a graph with $n$ vertices and $n$ edges must have at least one circuit.

WZ
Wen Zheng
Numerade Educator
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Problem 9

Let $h, k>0$ be fixed. Let $S=\{(i h, j k) \mid 1 \leq i \leq m, 1 \leq j \leq n\}$ be points in a rectangular planar grid. Show that the function space $\mathcal{F}(S)$ can be identified with the vector space of $m \times n$ matrices $\mathcal{M}_{m \times n}$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 9

A square matrix is called strictly lower triangular if all entries on or above the main diagonal are 0 . Prove that the space of strictly lower triangular matrices is a subspace of the vector space of all $n \times n$ matrices.

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Problem 9

Determine whether any of the following functions lies in the subspace spanned by $1, x$, $\sin x, \sin ^2 x$ : (a) $3-5 x,(b) x^2+\sin ^2 x$, (c) $\sin x-2 \cos x$, (d) $\cos ^2 x$, (e) $x \sin x,(f) e^x$.

Victor Salazar
Victor Salazar
Numerade Educator
01:18

Problem 9

(a) Prove that $1+t^2$ $\mathcal{P}^{(2)}$. (b) Find the coofdiPage $128 / /=702, t-7 t^2 \boldsymbol{Q}$ thit $\mathbf{t a s}$.
of quadratic polynomials

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 9

Let the square matrix $P$ be idempotent, meaning that $P^2=P$. (a) Prove that $\mathbf{w} \in \operatorname{img} P$ if and only if $P \mathbf{w}=\mathbf{w}$. (b) Show that img $P$ and $\operatorname{ker} P$ are complementary subspaces, as defined in Exercise 2.2.24, so every $\mathbf{v} \in \mathbb{R}^n$ can be uniquely written as $\mathbf{v}=\mathbf{w}+\mathbf{z}$ where $\mathbf{w} \in \operatorname{img} P, \mathbf{z} \in \operatorname{ker} P$.

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00:52

Problem 9

A connected graph is called a tree if it has no circuits. (a) Find the incidence matrix for each of the following directed trees:
(i)
(ii)
(iii)
(iv)
(b) Draw all distinct trees with 4 vertices. Assign a direction to the edges, and write down the corresponding incidence matrices. (c) Prove that a connected graph on $n$ vertices is a tree if and only if it has precisely $n-1$ edges.

WZ
Wen Zheng
Numerade Educator
13:49

Problem 10

The space $\mathbb{R}^{\infty}$ is defined as the set of all infinite real sequences $\mathbf{a}=\left(a_1, a_2, a_3, \ldots\right)$, where $a_i \in \mathbb{R}$. Define addition and scalar multiplication in such a way as to make $\mathbb{R}^{\infty}$ into a vector space. Explain why all the vector space axioms are valid.

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 10

Which of the following are subspaces of the vector space of $n \times n$ matrices $\mathcal{M}_{n \times n}$ ? The set of all (a) regular matrices; (b) nonsingular matrices; (c) singular matrices; (d) lower triangular matrices; (e) lower unitriangular matrices; $(f)$ diagonal matrices; $(g)$ symmetric matrices; $(h)$ skew-symmetric matrices.

Victor Salazar
Victor Salazar
Numerade Educator
04:39

Problem 10

Write the following trigonometric functions in phase-amplitude form:
(a) $\sin 3 x$,
(b) $\cos x-\sin x$,
(c) $3 \cos 2 x+4 \sin 2 x$,
(d) $\cos x \sin x$.

DN
Devon Nirschl
Numerade Educator

Problem 10

Find a basis for and the dimension of the span of
(a) $\left(\begin{array}{r}3 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}-6 \\ -2 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 3\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ -2\end{array}\right)$,
(c) $\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 2\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1 \\ 3\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ -3 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 1 \\ 1\end{array}\right)$

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Problem 10

Let $A$ be an $m \times n$ matrix. Suppose that $C=\left(\begin{array}{l}A \\ B\end{array}\right)$ is an $(m+k) \times n$ matrix whose first $m$ rows are the same Pagenose of $A$. 79 fove that ker $C \subseteq$ ker $A$. Thus, appending more rows cannot increase the size of a matrix's kernel. Give an example in which $\operatorname{ker} C \neq \operatorname{ker} A$.

Victor Salazar
Victor Salazar
Numerade Educator
16:34

Problem 10

A complete graph $G_n$ on $n$ vertices has one edge joining every distinct pair of vertices. (a) Draw $G_3, G_4$ and $G_5$. (b) Choose an orientation for each edge and write out the resulting incidence matrix of each digraph. (c) How many edges does $G_n$ have? $(d)$ How many independent circuits?

Chris Trentman
Chris Trentman
Numerade Educator
09:53

Problem 11

Prove the basic vector space properties $(i),(j),(k)$ following Definition 2.1 .

Anthony Ramos
Anthony Ramos
Numerade Educator
03:30

Problem 11

The trace of an $n \times n$ matrix $A \in \mathcal{M}_{n \times n}$ is defined to be the sum of its diagonal entries: $\operatorname{tr} A=a_{11}+a_{22}+\cdots+a_{n n}$. Prove that the set of trace zero matrices, $\operatorname{tr} A=0$, is a subspace of $\mathcal{M}_{n \times n}$.

Anas Venkitta
Anas Venkitta
Numerade Educator
01:10

Problem 11

(a) Prove that the set of solutions to the homogeneous ordinary differential equation $u^{\prime \prime}-4 u^{\prime}+3 u=0$ is a vector space. (b) Write the solution space as the span of a finite number of functions. (c) What is the minimal number of functions needed to span the solution space?

E R
E R
Numerade Educator
04:00

Problem 11

(a) Show that $1,1-t,(1-t)^2,(1-t)^3$ is a basis for $\mathcal{P}^{(3)}$.
(b) Write $p(t)=1+t^3$ in terms of the basis elements.

Ernest Castorena
Ernest Castorena
Numerade Educator
04:04

Problem 11

Let $A$ be an $m \times n$ matrix. Suppose that $C=(A B)$ is an $m \times(n+k)$ matrix whose first $n$ columns are the same as those of $A$. Prove that $\operatorname{img} C \supseteq \operatorname{img} A$. Thus, appending more columns cannot decrease the size of a matrix's image. Give an example in which $\operatorname{img} C \neq \operatorname{img} A$.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 11

The complete bipartite digraph $G_{m, n}$ is based on two disjoint sets of, respectively, $m$ and $n$ vertices. Each vertex in the first set is connected to each vertex in the second set by a single edge. (a) Draw $G_{2,3}, G_{2,4}$, and $G_{3,3}$. (b) Write the incidence matrix of each digraph. (c) How many edges does $G_{m, n}$ have? (d) How many independent circuits?

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Problem 12

Prove that a vector space has only one zero element $\mathbf{0}$.

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01:16

Problem 12

(a) Is the set of $n \times n$ matrices with $\operatorname{det} A=1$ a subspace of $\mathcal{M}_{n \times n}$ ?
(b) What about the matrices with $\operatorname{det} A=0$ ?

Sikandar Baig
Sikandar Baig
Numerade Educator
02:47

Problem 12

Explain why the functions $1, \cos x, \sin x \operatorname{span}$ the solution space to the third order ordinary differential equation $u^{\prime \prime \prime}+u^{\prime}=0$.

Aman Gupta
Aman Gupta
Numerade Educator
01:29

Problem 12

Let $\mathcal{P}^{(4)}$ denote the vector space consisting of all polynomials $p(x)$ of degree $\leq 4$.
(a) Are $x^3-3 x+1, x^4-6 x+3, x^4-2 x^3+1$ linearly independent elements of $\mathcal{P}^{(4)}$ ?
(b) What is the dimension of the subspace of $\mathcal{P}^{(4)}$ they span?

Cory Glover
Cory Glover
Numerade Educator
01:08

Problem 12

Find the solution $\mathbf{x}_1^{\star}$ to the system $\left(\begin{array}{rr}1 & 2 \\ -3 & -4\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}1 \\ 0\end{array}\right)$, and the solution $\mathbf{x}_2^{\star}$ to $\left(\begin{array}{rr}1 & 2 \\ -3 & -4\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}0 \\ 1\end{array}\right)$. Express the solution to $\left(\begin{array}{rr}1 & 2 \\ -3 & -4\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}1 \\ 4\end{array}\right)$ as a linear combination of $\mathbf{x}_1^{\star}$ and $\mathbf{x}_2^{\star}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 12

(a) Construct the incidence matrix $A$ for the disconnected digraph $D$ in the figure. (b) Verify that $\operatorname{dim} \operatorname{ker} A=3$, which is the same as the number of connected components, meaning the maximal connected subgraphs in $D$. (c) Can you assign an interpretation to your basis for ker $A$ ? (d) Try proving the general statement that $\operatorname{dim} \operatorname{ker} A$ equals the number of connected components in the digraph $D$.

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01:29

Problem 13

Suppose that $V$ and $W$ are vector spaces. The Cartesian product space, denoted by $V \times W$, is defined as the set of all ordered pairs $(\mathbf{v}, \mathbf{w})$, where $\mathbf{v} \in V, \mathbf{w} \in W$, with vector addition $(\mathbf{v}, \mathbf{w})+(\widehat{\mathbf{v}}, \widehat{\mathbf{w}})=(\mathbf{v}+\widehat{\mathbf{v}}, \mathbf{w}+\widehat{\mathbf{w}})$ and scalar multiplication $c(\mathbf{v}, \mathbf{w})=(c \mathbf{v}, c \mathbf{w})$.
(a) Prove that $V \times W$ is a vector space. (b) Explain why $\mathbb{R} \times \mathbb{R}$ is the same as $\mathbb{R}^2$.
(c) More generally, explain why $\mathbb{R}^m \times \mathbb{R}^n$ is the same as $\mathbb{R}^{m+n}$.

Runpeng Li
Runpeng Li
Numerade Educator
01:43

Problem 13

Let $V=\mathrm{C}^0(\mathbb{R})$ be the vector space consisting of all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Explain why the set of all functions such that $f(1)=0$ is a subspace, but the set of functions such that $f(0)=1$ is not. For which values of $a, b$ does the set of functions such that $f(a)=b$ form a subspace?

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:07

Problem 13

Find a finite set of real functions that spans the solution space to the following homogeneous ordinary differential equations: (a) $u^{\prime}-2 u=0$, (b) $u^{\prime \prime}+4 u=0$, (c) $u^{\prime \prime}-3 u^{\prime}=0$,
(d) $u^{\prime \prime}+u^{\prime}+u=0$,
(e) $u^{\prime \prime \prime}-5 u^{\prime \prime}=0$,
(f) $u^{(4)}+u=0$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
04:05

Problem 13

Let $S=\left\{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}\right\}$. (a) Show that the sample vectors corresponding to the functions $1, \cos \pi x, \cos 2 \pi x$, and $\cos 3 \pi x$ form a basis for the vector space of all sample functions on S. (b) Write the sampled version of the function $f(x)=x$ in terms of this basis.

Jimmy Yao
Jimmy Yao
Numerade Educator
01:45

Problem 13

Let $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 5 & -1 \\ 1 & 3 & 2\end{array}\right)$. Given that $\mathbf{x}_1^{\star}=\left(\begin{array}{r}5 \\ -1 \\ 2\end{array}\right)$ solves $A \mathbf{x}=\mathbf{b}_1=\left(\begin{array}{l}1 \\ 3 \\ 6\end{array}\right)$ and $\mathbf{x}_2^{\star}=\left(\begin{array}{r}-11 \\ 5 \\ -1\end{array}\right)$ solves $A \mathbf{x}=\mathbf{b}_2=\left(\begin{array}{l}0 \\ 4 \\ 2\end{array}\right)$, find a solution to $A \mathbf{x}=2 \mathbf{b}_1+\mathbf{b}_2=\left(\begin{array}{r}2 \\ 10 \\ 14\end{array}\right)$.

Abhijith V
Abhijith V
Numerade Educator
05:07

Problem 13

How does altering the direction of the edges of a digraph affect its incidence matrix? The cokernel of its incidence matrix? Can you realize this operation by matrix multiplication?

Chris Trentman
Chris Trentman
Numerade Educator

Problem 14

2.1.14. Use Exercise 2.1.13 to show that the space of pairs $(f(x), a)$, where $f$ is a continuous scalar function and $a$ is a real number, is a vector space. What is the zero element? Be precise! Write out the laws of vector addition and scalar multiplication.

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06:59

Problem 14

Which of the following are vector spaces? Justify your answer! (a) The set of all row vectors of the form $(a, 3 a)$. (b) The set of all vectors of the form $(a, a+1)$. (c) The set of all continuous functions for which $f(-1)=0$. (d) The set of all periodic functions of period 1, i.e., $f(x+1)=f(x)$. (e) The set of all non-negative functions: $f(x) \geq 0$.
(f) The set of all even polynomials: $p(x)=p(-x)$. (g) The set of all polynomials $p(x)$ that have $x-1$ as a factor. (h) The set of all quadratic forms $q(x, y)=a x^2+b x y+c y^2$.

Noor Aldeen Almusleh
Noor Aldeen Almusleh
Numerade Educator
03:19

Problem 14

Consider the boundary value problem $u^{\prime \prime}+4 u=0,0 \leq x \leq \pi, u(0)=0, u(\pi)=0$.
(a) Prove, without solving, that the set of solutions forms a vector space.
(b) Write this space as the span of one or more functions. Hint: First solve the differential equation; then find out which solutions satisfy the boundary conditions.

Madi Sousa
Madi Sousa
Numerade Educator
01:44

Problem 14

(a) Prove that the vector space of all $2 \times 2$ matrices is a four-dimensional vector space by exhibiting a basis. (b) Generalize your result and prove that the vector space $\mathcal{M}_{m \times n}$ consisting of all $m \times n$ matrices has dimension $m n$.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:08

Problem 14

(a) Show that $\mathbf{x}_1^{\star}=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ and $\mathbf{x}_2^{\star}=\left(\begin{array}{r}-3 \\ 3 \\ -2\end{array}\right)$ are particular solutions to the system $\left(\begin{array}{rrr}2 & -1 & -5 \\ 1 & -4 & -6 \\ 3 & 2 & -4\end{array}\right) \mathbf{x}=\left(\begin{array}{r}1 \\ -3 \\ 5\end{array}\right)$. (b) Find the general solution.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
11:05

Problem 14

(a) Explain why two digraphs are equivalent under relabeling of vertices and edges if and only if their incidence matrices satisfy $P A Q=B$, where $P, Q$ are permutation matrices. (b) Decide which of the following incidence matrices produce the equivalent digraphs:
(i) $\left(\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right)$,
(ii)
(ii) $\left(\begin{array}{rrrr}0 & -1 & 1 & 0 \\ -1 & 0 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & -1 & 0 & 1\end{array}\right)$,
(iii) $\left(\begin{array}{rrrr}1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 1 & 0 & -1 & 0 \\ 0 & 0 & -1 & 1\end{array}\right)$,
(iv) $\left(\begin{array}{rrrr}1 & -1 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 1\end{array}\right)$,
(v) $\left(\begin{array}{rrrr}1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 1 \\ 0 & -1 & 1 & 0 \\ 1 & -1 & 0 & 0\end{array}\right)$,
(vi)
$$
\left(\begin{array}{rrrr}
1 & -1 & 0 & 0 \\
0 & -1 & 1 & 0 \\
0 & -1 & 0 & 1 \\
-1 & 0 & 1 & 0
\end{array}\right)
$$
(c) How are the cokernels of equivalent incidence matrices related?

Chris Trentman
Chris Trentman
Numerade Educator
05:36

Problem 15

Determine which of the following conditions describe subspaces of the vector space $\mathrm{C}^1$ consisting of all continuously differentiable scalar functions $f(x)$.
(a) $f(2)=f(3),(b) f^{\prime}(2)=f(3),(c) f^{\prime}(x)+f(x)=0,(d) \quad f(2-x)=f(x)$,
(e) $f(x+2)=f(x)+2$, (f) $f(-x)=e^x f(x)$. (g) $f(x)=a+b|x|$ for some $a, b \in \mathbb{R}$,

Nasheed Jafri
Nasheed Jafri
Numerade Educator
01:20

Problem 15

Which of the following functions lie in the span of the vector-valued functions
$$
\mathbf{f}_1(x)=\left(\begin{array}{l}
1 \\
x
\end{array}\right), \quad \mathbf{f}_2(x)=\left(\begin{array}{c}
x \\
1
\end{array}\right), \quad \mathbf{f}_3(x)=\left(\begin{array}{c}
x \\
2 x
\end{array}\right) ?
$$
(a) $\left(\begin{array}{l}2 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{c}1-2 x \\ 1-x\end{array}\right)$,
(c) $\left(\begin{array}{c}1-2 x \\ -1-x\end{array}\right)$,
(d) $\left(\begin{array}{c}1+x^2 \\ 1-x^2\end{array}\right)$,
(e) $\left(\begin{array}{c}2-x \\ 0\end{array}\right)$

Nick Johnson
Nick Johnson
Numerade Educator
10:46

Problem 15

Determine all values of the scalar $k$ for which the following four matrices form a basis for $\mathcal{M}_{2 \times 2}: A_1=\left(\begin{array}{rr}1 & -1 \\ 0 & 0\end{array}\right), A_2=\left(\begin{array}{rr}k & -3 \\ 1 & 0\end{array}\right), A_3=\left(\begin{array}{rr}1 & 0 \\ -k & 2\end{array}\right), A_4=\left(\begin{array}{rr}0 & k \\ -1 & -2\end{array}\right)$.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
01:47

Problem 15

A physical apparatus moves 2 meters under a force of 4 newtons. Assuming linearity, how far will it move under a force of 10 newtons?

Raushan Kumar
Raushan Kumar
Numerade Educator
00:24

Problem 15

True or false: If $A$ and $B$ are incidence matrices of the same size and coker $A=\operatorname{coker} B$, then the corresponding digraphs are equivalent.

AG
Ankit Gupta
Numerade Educator

Problem 16

Let $V=\mathrm{C}^0[a, b]$ be the vector space consisting of all functions $f(t)$ that are defined and continuous on the interval $0 \leq t \leq 1$. Which of the following conditions define subspaces of $V$ ? Explain your answer. (a) $f(0)=0$, (b) $f(0)=2 f(1),(c) f(0) f(1)=1$,
(d) $f(0)=0$ or $f(1)=0$,
(e) $f(1-t)=-t f(t)$,
(f) $f(1-t)=1-f(t)$,
(g) $f\left(\frac{1}{2}\right)=\int_0^1 f(t) d t$,
(h) $\int_0^1(t-1) f(t) d t=0$,
(i) $\int_0^t f(s) \sin s d s=\sin t$.

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00:51

Problem 16

True or false: The zero vector belongs to the span of any collection of vectors.

Monica Miller
Monica Miller
Numerade Educator
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Problem 16

Prove that the space of diagonal $n \times n$ matrices is an $n$-dimensional vector space.

Nick Johnson
Nick Johnson
Numerade Educator
02:24

Problem 16

Applying a unit external force in the horizontal direction moves a mass 3 units to the right, while applying a unit force in the vertical direction moves it up 2 units. Assuming linearity, where will the mass move under the applied force $\mathbf{f}=(2,-3)^T$ ?

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
01:38

Problem 16

(a) Explain why the incidence matrix for a disconnected graph can be written in block diagonal matrix form $A=\left(\begin{array}{ll}B & \mathrm{O} \\ \mathrm{O} & C\end{array}\right)$ under an appropriate labeling of the vertices.
(b) Show how to label the vertices of the digraph in Exercise 2.6.3e so that its incidence matrix is in block form.

Chris Trentman
Chris Trentman
Numerade Educator
03:47

Problem 17

Prove that the set of solutions to the second order ordinary differential equation $u^{\prime \prime}=x u$ is a vector space.

Uma Kumari
Uma Kumari
Numerade Educator
04:03

Problem 17

Prove or give a counter-example: if $\mathbf{z}$ is a linear combination of $\mathbf{u}, \mathbf{v}, \mathbf{w}$, then $\mathbf{w}$ is a linear combination of $\mathbf{u}, \mathbf{v}, \mathbf{z}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
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Problem 17

(a) Find a basis for and the dimension of the space of upper triangular $2 \times 2$ matrices.
(b) Can you generalize your result to upper triangular $n \times n$ matrices?

Victor Salazar
Victor Salazar
Numerade Educator
03:24

Problem 17

Suppose $\mathbf{x}_1^{\star}$ and $\mathbf{x}_2^{\star}$ are both solutions to $A \mathbf{x}=\mathbf{b}$. List all linear combinations of $\mathbf{x}_1^{\star}$ and $\mathbf{x}_2^{\star}$ that solve the system.

Adhish Rele
Adhish Rele
Numerade Educator
04:09

Problem 18

Show that the set of solutions to $u^{\prime \prime}=x+u$ does not form a vector space.

oh
Oday Hazaimah
Northern Illinois University
08:32

Problem 18

Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_m$ span $V$. Let $\mathbf{v}_{m+1}, \ldots, \mathbf{v}_n \in V$ be any other elements. Prove that the combined collection $\mathbf{v}_1, \ldots, \mathbf{v}_n$ also spans $V$.

WM
William Mead
Numerade Educator
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Problem 18

(a) What is the dimension of the vector space of $2 \times 2$ symmetric matrices? Of skewsymmetric matrices? (b) Generalize to the $3 \times 3$ case. (c) What about $n \times n$ matrices?

Victor Salazar
Victor Salazar
Numerade Educator
02:45

Problem 18

Let $A$ be a nonsingular $m \times m$ matrix. (a) Explain in detail why the solutions $\mathbf{x}_1^{\star}, \ldots, \mathbf{x}_m^{\star}$ to the systems $(2.34)$ are the columns of the matrix inverse $A^{-1}$.
(b) Illustrate your argument in the case $A=\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 1 & 3 \\ 1 & 0 & 1\end{array}\right)$.

Prashant Bana
Prashant Bana
Numerade Educator

Problem 19

(a) Prove that $\mathrm{C}^1\left([a, b], \mathbb{R}^2\right)$, which is the space of continuously differentiable parameterized plane curves $\mathbf{f}:[a, b] \rightarrow \mathbb{R}^2$, is a vector space.
(b) Is the subset consisting of all curves that go through the origin a subspace?

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Problem 19

(a) Show that if $\mathbf{v}$ is a linear combination of $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and each $\mathbf{v}_j$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$, then $\mathbf{v}$ is a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_n$.
(b) Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_m$ span $V$. Let $\mathbf{w}_1, \ldots, \mathbf{w}_m \in V$ be any other elements. Suppose that each $\mathbf{v}_i$ can be written as a linear combination of $\mathbf{w}_1, \ldots, \mathbf{w}_m$. Prove that $\mathbf{w}_1, \ldots, \mathbf{w}_m$ also $\operatorname{span} V$.

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02:32

Problem 19

A matrix is said to be a semi-magic square if its row sums and column sums (i.e., the sum of entries in an individual row or column) all add up to the same number. An example is $\left(\begin{array}{lll}8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2\end{array}\right)$, whose row and column sums are all equal to 15 . (a) Explain why the set of all semi-magic squares is a subspace of the vector space of $3 \times 3$ matrices. (b) Prove that the $3 \times 3$ permutation matrices (1.30) span the space of semi-magic squares. What is its dimension? (c) A magic square also has the diagonal and anti-diagonal (running from top right to bottom left) add up to the common row and column sum; the preceding $3 \times 3$ example is magic. Does the set of $3 \times 3$ magic squares form a vector space? If so, what is its dimension? (d) Write down a formula for all $3 \times 3$ magic squares.

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:32

Problem 19

True or false: If $\mathbf{x}_1^{\star}$ solves $A \mathbf{x}=\mathbf{c}$, and $\mathbf{x}_2^{\star}$ solves $B \mathbf{x}=\mathbf{d}$, then $\mathbf{x}^{\star}=\mathbf{x}_1^{\star}+\mathbf{x}_2^{\star}$ solves $(A+B) \mathbf{x}=\mathbf{c}+\mathbf{d}$.
2.5.20. Under what conditions on the coefficient matrix $A$ will the systems in (2.34) all have a solution?

Victor Salazar
Victor Salazar
Numerade Educator
00:09

Problem 20

A planar vector field $\mathbf{v}(x, y)=(u(x, y), v(x, y))^T$ is called irrotational if it has zero divergence: $\nabla \cdot \mathbf{v}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \equiv 0$. Prove that the set of all irrotational vector fields is a subspace of the space of all planar vector fields.

Frank Lin
Frank Lin
Numerade Educator
02:27

Problem 20

The span of an infinite collection $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \ldots \in V$ of vector space elements is defined as the set of all finite linear combinations $\sum_{i=1}^n c_i \mathbf{v}_i$, where $n<\infty$ is finite but arbitrary.
(a) Prove that the span defines a subspace of the vector space $V$.
(b) What is the span of the monomials $1, x, x^2, x^3, \ldots$ ?

Harshita Goel
Harshita Goel
Numerade Educator
08:16

Problem 20

(a) Prove that if $\mathbf{v}_1, \ldots, \mathbf{v}_m$ forms a basis for $V \subsetneq \mathbb{R}^n$, then $m<n$. (b) Under the hypothesis of part (a), prove that there exist vectors $\mathbf{v}_{m+1}, \ldots, \mathbf{v}_n \in \mathbb{R}^n \backslash V$ such that the complete collection $\mathbf{v}_1, \ldots, \mathbf{v}_n$ forms a basis for $\mathbb{R}^n$. (c) Illustrate by constructing bases of $\mathbb{R}^3$ that include $(i)$ the basis $\left(1,1, \frac{1}{2}\right)^T$ of the line $x=y=2 z ;(i i)$ the basis $(1,0,-1)^T$, $(0,1,-2)^T$ of the plane $x+2 y+z=0$.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:34

Problem 20

Under what conditions on the coefficient matrix $A$ will the systems in (2.34) all have a solution?

Jingyun Wang
Jingyun Wang
Numerade Educator
02:01

Problem 21

Let $C \subset \mathbb{R}^{\infty}$ denote the set of all convergent sequences of real numbers, where $\mathbb{R}^{\infty}$ was defined in Exercise 2.2.21. Is $C$ a subspace?

Harshita Goel
Harshita Goel
Numerade Educator
01:35

Problem 21

Determine whether the given vectors are linearly independent or linearly dependent:
(a) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{l}1 \\ 3\end{array}\right),\left(\begin{array}{l}-2 \\ -6\end{array}\right)$,
(c) $\left(\begin{array}{l}2 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 3\end{array}\right),\left(\begin{array}{l}5 \\ 2\end{array}\right)$,
(d) $\left(\begin{array}{r}1 \\ 3 \\ -2\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}3 \\ -1 \\ 2\end{array}\right)$,
(f) $\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -3 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 4\end{array}\right)$,
(g) $\left(\begin{array}{r}4 \\ 2 \\ 0 \\ -6\end{array}\right),\left(\begin{array}{r}-6 \\ -3 \\ 0 \\ 9\end{array}\right)$,
(h) $\left(\begin{array}{r}2 \\ 1 \\ -1 \\ 3\end{array}\right),\left(\begin{array}{r}-1 \\ 3 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}5 \\ 1 \\ 2 \\ -3\end{array}\right)$,
(i) $\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 2 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right)$,

Shima Shaw
Shima Shaw
Numerade Educator
09:00

Problem 21

Suppose that $\mathbf{v}_1, \ldots, \mathbf{v}_n$ form a basis for $\mathbb{R}^n$. Let $A$ be a nonsingular matrix. Prove that $A \mathbf{v}_1, \ldots, A \mathbf{v}_n$ also form a basis for $\mathbb{R}^n$. What is this basis if you start with the standard basis: $\mathbf{v}_i=\mathbf{e}_i$ ?

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 21

For each of the following matrices find bases for the (i) image, (ii) coimage, (iii) kernel, and (iv) cokernel.
(a) $\left(\begin{array}{ll}1 & -3 \\ 2 & -6\end{array}\right)$,
(b) $\left(\begin{array}{rrr}0 & 0 & -8 \\ 1 & 2 & -1 \\ 2 & 4 & 6\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}1 & 1 & 2 & 1 \\ 1 & 0 & -1 & 3 \\ 2 & 3 & 7 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrrrr}1 & -3 & 2 & 2 & 1 \\ 0 & 3 & -6 & 0 & -2 \\ 2 & -3 & -2 & 4 & 0 \\ 3 & -3 & -6 & 6 & 3 \\ 1 & 0 & -4 & 2 & 3\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 22

Show that if $W$ and $Z$ are subspaces of $V$, then (a) their intersection $W \cap Z$ is a subspace of $V,(b)$ their sum $W+Z=\{\mathbf{w}+\mathbf{z} \mid \mathbf{w} \in W, \mathbf{z} \in Z\}$ is also a subspace, but (c) their union $W \cup Z$ is not a subspace of $V$, unless $W \subset Z$ or $Z \subset W$.

Victor Salazar
Victor Salazar
Numerade Educator
02:32

Problem 22

(a) Show that the vectors $\left(\begin{array}{l}1 \\ 0 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 3 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -2 \\ 1 \\ -1\end{array}\right)$ are linearly independent. (b) Whi $)$ of the following vectors are in their span? (i) $\left(\begin{array}{l}1 \\ 1 \\ 2 \\ 1\end{array}\right)$, (ii) $\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 0\end{array}\right)$, (iii) $\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 0\end{array}\right)$, (iv) $\left(\begin{array}{l}0 \\ 0 \\ 0 \\ 0\end{array}\right)$.
(c) Suppose $\mathbf{b}=(a, b, c, d)^T$ lies in their span. What conditions must $a, b, c, d$ satisfy?

Donald Albin
Donald Albin
Numerade Educator
01:38

Problem 22

Show that if $\mathbf{v}_1, \ldots, \mathbf{v}_n$ span $V \neq\{\mathbf{0}\}$, then one can choose a subset $\mathbf{v}_{i_1}, \ldots, \mathbf{v}_{i_m}$ that forms a basis of $V$. Thus, $\operatorname{dim} V=m \leq n$. Under what conditions is $\operatorname{dim} V=n$ ?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 22

Find a set of columns of the matrix $\left(\begin{array}{rrrrr}-1 & 2 & 0 & -3 & 5 \\ 2 & -4 & 1 & 1 & -4 \\ -3 & 6 & 2 & 0 & 8\end{array}\right)$ that form a basis for its image. Then express each column as a linear combination of the basis columns.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 23

Let $V$ be a vector space. Prove that the intersection $\cap W_i$ of any collection (finite or infinite) of subspaces $W_i \subset V$ is a subspace.

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08:26

Problem 23

(a) Show that the vectors $\left(\begin{array}{l}1 \\ 1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ 1 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 0 \\ -1\end{array}\right)$ are linearly independent.
(b) Show that they also span $\mathbb{R}^4$. (c) Write $(1,0,0,1)^T$ as a linear combination of them.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
07:23

Problem 23

Prove that if $\mathbf{v}_1, \ldots, \mathbf{v}_n$ are a basis of $V$, then every subset thereof, e.g., $\mathbf{v}_{i_1}, \ldots, \mathbf{v}_{i_k}$, is linearly independent.

Anthony Ramos
Anthony Ramos
Numerade Educator

Problem 23

For each of the following matrices $A$ : (a) Determine the rank and the dimensions of the four fundamental subspaces. (b) Find bases for both the kernel and cokernel. (c) Find explicit conditions on vectors $\mathbf{b}$ that guarantee that the system $A \mathbf{x}=\mathbf{b}$ has a solution. (d) Write down a specific nonzero vector $\mathbf{b}$ that satisfies your conditions, and then find all possible solutions $\mathbf{x}$.
(i) $\left(\begin{array}{rr}1 & 2 \\ -2 & -4\end{array}\right)$,
(ii) $\left(\begin{array}{rrr}3 & -1 & -2 \\ -6 & 2 & 4\end{array}\right)$,
(iii) $\left(\begin{array}{rr}1 & 5 \\ -2 & 3 \\ 2 & 7\end{array}\right)$,
(iv) $\left(\begin{array}{rrr}2 & -5 & -1 \\ 1 & -6 & -4 \\ 3 & -4 & 2\end{array}\right)$,
(v) $\left(\begin{array}{rrr}2 & 5 & 7 \\ 6 & 13 & 19 \\ 3 & 8 & 11 \\ 1 & 2 & 3\end{array}\right)$,
(vi) $\left(\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 1 \\ 1 & -2 & 2 & 7 \\ 3 & 6 & 5 & -2\end{array}\right)$,
(vii)
$\left(\begin{array}{rrrrr}2 & 4 & 0 & -6 & 0 \\ 1 & 2 & 3 & 15 & 0 \\ 3 & 6 & -1 & 15 & 5 \\ -3 & -6 & 2 & 21 & -6\end{array}\right)$

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Problem 24

Let $W \subset V$ be a subspace. A subspace $Z \subset V$ is called a complementary subspace to $W$ if (i) $W \cap Z=\{0\}$, and (ii) $W+Z=V$, i.e., every $\mathbf{v} \in V$ can be written as $\mathbf{v}=\mathbf{w}+\mathbf{z}$ for $\mathbf{w} \in W$ and $\mathbf{z} \in Z$. (a) Show that the $x$ - and $y$-axes are complementary subspaces of $\mathbb{R}^2$. (b) Show that the lines $x=y$ and $x=3 y$ are complementary subspaces of $\mathbb{R}^2$. (c) Show that the line $(a, 2 a, 3 a)^T$ and the plane $x+2 y+3 z=0$ are complementary subspaces of $\mathbb{R}^3$. (d) Prove that if $\mathbf{v}=\mathbf{w}+\mathbf{z}$, then $\mathbf{w} \in W$ and $\mathbf{z} \in Z$ are uniquely determined.

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02:25

Problem 24

Determine whether the given row vectors are linearly independent or linearly dependent:
(a) $(2,1),(-1,3),(5,2), \quad$ (b) $(1,2,-1),(2,4,-2), \quad$ (c) $(1,2,3),(1,4,8),(1,5,7)$,
(d) $(1,1,0),(1,0,3),(2,2,1),(1,3,4), \quad($ e) $(1,2,0,3),(-3,-1,2,-2),(3,-4,-4,5)$,
(f) $(2,1,-1,3),(-1,3,1,0),(5,1,2,-3)$.

Shima Shaw
Shima Shaw
Numerade Educator
01:01

Problem 24

Show, by example, how the uniqueness result in Lemma 2.34 fails if one has a linearly dependent set of vectors.

Shima Shaw
Shima Shaw
Numerade Educator

Problem 24

Find the dimension of and a basis for the subspace spanned by the following sets of vectors. Hint: First identify the subspace with the image of a certain matrix.
(a)
$$
\left(\begin{array}{r}
1 \\
2 \\
-1
\end{array}\right),\left(\begin{array}{l}
2 \\
2 \\
0
\end{array}\right) \text {, }
$$
(b)
$$
\left(\begin{array}{r}
1 \\
1 \\
-1
\end{array}\right),\left(\begin{array}{r}
2 \\
2 \\
-2
\end{array}\right),\left(\begin{array}{r}
-3 \\
-3 \\
3
\end{array}\right)
$$
(c) $\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 2 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ 3 \\ -3\end{array}\right)$,
(d) $\left(\begin{array}{r}1 \\ 0 \\ -3 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ 1 \\ 2 \\ -3\end{array}\right),\left(\begin{array}{r}-3 \\ -4 \\ 1 \\ 6\end{array}\right),\left(\begin{array}{r}1 \\ -3 \\ -8 \\ 7\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ -6 \\ 9\end{array}\right),(e)\left(\begin{array}{r}1 \\ 1 \\ -1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 2 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}3 \\ 0 \\ 1 \\ 3 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -3 \\ 4 \\ 0 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ 3 \\ -1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 3 \\ 2 \\ 0\end{array}\right)$.

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Problem 25

(a) Show that $V_0=\{(\mathbf{v}, \mathbf{0}) \mid \mathbf{v} \in V\}$ and $W_0=\{(\mathbf{0}, \mathbf{w}) \mid \mathbf{w} \in W\}$ are complementary subspaces, as in Exercise 2.2.24, of the Cartesian product space $V \times W$, as defined in Exercise 2.1.13. (b) Prove that the diagonal $D=\{(\mathbf{v}, \mathbf{v})\}$ and the anti-diagonal $A=\{(\mathbf{v}-\mathbf{v})\}$ are complementary subspaces of $V \times V$.

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01:32

Problem 25

True or false: The six $3 \times 3$ permutation matrices (1.30) are linearly independent.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 25

Let $W \subset V$ be a subspace. (a) Prove that $\operatorname{dim} W \leq \operatorname{dim} V$.
(b) Prove that if $\operatorname{dim} W=\operatorname{dim} V=n<\infty$, then $W=V$. Equivalently, if $W \subsetneq V$ is a proper subspace of a finite-dimensional vector space, then $\operatorname{dim} W<\operatorname{dim} V$.
(c) Give an example in which the result is false if $\operatorname{dim} V=\infty$.

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02:25

Problem 25

Show that the set of all vectors $\mathbf{v}=(a-3 b, a+2 c+4 d, b+3 c-d, c-d)^T$, where $a, b, c, d$ are real numbers, forms a subspace of $\mathbb{R}^4$, and find its dimension.

VU
Viswesh Uppalapati
Numerade Educator
02:35

Problem 26

Show that the set of skew-symmetric $n \times n$ matrices forms a complementary subspace to the set of symmetric $n \times n$ matrices. Explain why this implies that every square matrix can be uniquely written as the sum of a symmetric and a skew-symmetric matrix.

Urvashi Arora
Urvashi Arora
Numerade Educator
00:49

Problem 26

True or false: A set of vectors is linearly dependent if the zero vector belongs to their span.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:49

Problem 26

Let $W, Z \subset V$ be complementary subspaces in a finite-dimensional vector space $V$, as in Exercise

Nick Johnson
Nick Johnson
Numerade Educator
05:49

Problem 26

Find a basis of the solution space of the following homogeneous linear systems.
(a)
$$
\begin{aligned}
x_1-2 x_3 & =0, \\
x_2+x_4 & =0 .
\end{aligned}
$$
(b)
$$
\begin{array}{r}
2 x_1+x_2-3 x_3+x_4=0, \\
2 x_1-x_2-x_3-x_4=0 .
\end{array}
$$
$$
\begin{array}{r}
x_1-x_2-2 x_3+4 x_4=0, \\
2 x_1+x_2-x_4=0, \\
-2 x_1+2 x_3-2 x_4=0 .
\end{array}
$$

Barsha Rana
Barsha Rana
Numerade Educator
View

Problem 27

(a) Show that the set of even functions, $f(-x)=f(x)$, is a subspace of the vector space of all functions $\mathcal{F}(\mathbb{R})$. (b) Show that the set of odd functions, $g(-x)=-g(x)$, forms a complementary subspace, as defined in Exercise 2.2.24. (c) Explain why every function can be uniquely written as the sum of an even function and an odd function.

Carson Merrill
Carson Merrill
Numerade Educator
02:51

Problem 27

Does a single vector ever define a linearly dependent set?

ET
Ed Tam
Numerade Educator
02:47

Problem 27

Let $V$ be a finite-dimensional vector space and $W \subset V$ a subspace. Prove that the quotient space, as defined in Exercise 2.2.29, has dimension $\operatorname{dim}(V / W)=\operatorname{dim} V-\operatorname{dim} W$.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 27

Find bases for the image and coimage of $\left(\begin{array}{rrr}1 & -3 & 0 \\ 2 & -6 & 4 \\ -3 & 9 & 1\end{array}\right)$. Make sure they have the same number of elements. Then write each row and column as a linear combination of the appropriate basis vectors.

Nick Johnson
Nick Johnson
Numerade Educator
02:01

Problem 28

Let $V$ be a vector space. A subset of the form $A=\{\mathbf{w}+\mathbf{a} \mid \mathbf{w} \in W\}$, where $W \subset V$ is a subspace and $\mathbf{a} \in V$ is a fixed vector, is known as an affine subspace of $V$. (a) Show that an affine subspace $A \subset V$ is a genuine subspace if and only if $\mathbf{a} \in W$. (b) Draw the affine subspaces $A \subset \mathbb{R}^2$ when (i) $W$ is the $x$-axis and $\mathbf{a}=(2,1)^T$, (ii) $W$ is the line $y=\frac{3}{2} x$ and $\mathbf{a}=(1,1)^T$, (iii) $W$ is the line $\left\{(t,-t)^T \mid t \in \mathbb{R}\right\}$, and $\mathbf{a}=(2,-2)^T$. (c) Prove that every affine subspace $A \subset \mathbb{R}^2$ is either a point, a line, or all of $\mathbb{R}^2$. (d) Show that the plane $x-2 y+3 z=1$ is an affine subspace of $\mathbb{R}^3$. (e) Show that the set of all polynomials such that $p(0)=1$ is an affine subspace of $\mathcal{P}^{(n)}$.

Harshita Goel
Harshita Goel
Numerade Educator
03:22

Problem 28

Let $\mathbf{x}$ and $\mathbf{y}$ be linearly independent elements of a vector space $V$. Show that $\mathbf{u}=a \mathbf{x}+b \mathbf{y}$, and $\mathbf{v}=c \mathbf{x}+d \mathbf{y}$ are linearly independent if and only if $a d-b c \neq 0$. Is the entire collection $\mathbf{x}, \mathbf{y}, \mathbf{u}, \mathbf{v}$ linearly independent?

WM
William Mead
Numerade Educator

Problem 28

Let $f_1(x), \ldots, f_n(x)$ be scalar functions. Suppose that every set of sample points $x_1, \ldots, x_m \in \mathbb{R}$, for all finite $m \geq 1$, leads to linearly dependent sample vectors $\mathbf{f}_1, \ldots, \mathbf{f}_n \in \mathbb{R}^m$. Prove that $f_1(x), \ldots, f_n(x)$ are linearly dependent functions.
Hint: Given sample points $x_1, \ldots, x_m$, let $V_{x_1, \ldots, x_m} \subset \mathbb{R}^n$ be the subspace consisting of all vectors $\mathbf{c}=\left(c_1, c_2, \ldots, c_n\right)^T$ such that $c_1 \mathbf{f}_1+\cdots+c_n \mathbf{f}_n=\mathbf{0}$. First, show that one can select sample points $x_1, x_2, x_3, \ldots$ such that $\mathbb{R}^n \supsetneq V_{x_1} \supsetneq V_{x_1, x_2} \supsetneq \cdots$. Then, apply Exercise 2.4 .25 to conclude that $V_{x_1, \ldots, x_n}=\{0\}$.

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04:45

Problem 28

Find bases for the image of $\left(\begin{array}{rrr}1 & 2 & -1 \\ 0 & 3 & -3 \\ 2 & -4 & 6 \\ 1 & 5 & -4\end{array}\right)$ using both of the indicated methods.
Demonstrate that they are indeed both bases for the same subspace by showing how to write each basis in terms of the other.

Ernest Castorena
Ernest Castorena
Numerade Educator
02:27

Problem 29

Quotient spaces: Let $V$ be a vector space and $W \subset V$ a subspace. We say that two vectors $\mathbf{u}, \mathbf{v} \in V$ are equivalent modulo $W$ if $\mathbf{u}-\mathbf{v} \in W$. (a) Show that this defines an equivalence relation, written $\mathbf{u} \sim_W \mathbf{v}$ on $V$, i.e., $(i) \mathbf{v} \sim_W \mathbf{v}$ for every $\mathbf{v}$; (ii) if $\mathbf{u} \sim_W \mathbf{v}$, then $\mathbf{v} \sim_W \mathbf{u}$; and (iii) if, in addition, $\mathbf{v} \sim_W \mathbf{z}$, then $\mathbf{u} \sim_W \mathbf{z}$. (b) The equivalence class of a vector $\mathbf{u} \in V$ is defined as the set of all equivalent vectors, written $[\mathbf{u}]_W=\left\{\mathbf{v} \in V \mid \mathbf{v} \sim_W \mathbf{u}\right\}$. Show that $[\mathbf{0}]_W=W$. (c) Let $V=\mathbb{R}^2$ and $W=\left\{(x, y)^T \mid x=2 y\right\}$. Sketch a picture of several equivalence classes as subsets of $\mathbb{R}^2$. (d) Show that each equivalence class $[\mathbf{u}]_W$ for $\mathbf{u} \in V$ is an affine subspace of $V$, as in Exercise 2.2.28. (e) Prove that the set of equivalence classes, called the quotient space and denoted by $V / W=\{[\mathbf{u}] \mid \mathbf{u} \in V\}$, forms a vector space under the operations of addition, $[\mathbf{u}]_W+[\mathbf{v}]_W=[\mathbf{u}+\mathbf{v}]_W$, and scalar multiplication, $c[\mathbf{u}]_W=[c \mathbf{u}]_W$. What is the zero element? Thus, you first need to prove that these operations are well defined, and then demonstrate the vector space axioms.

AP
Andreas Papavassiliou
Numerade Educator

Problem 29

Prove or give a counterexample to the following statement: If $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are elements of a vector space $V$ that do not span $V$, then $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent.

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Problem 29

Show that $\mathbf{v}_1=(1,2,0,-1)^T, \mathbf{v}_2=(-3,1,1,-1)^T, \mathbf{v}_3=(2,0,-4,3)^T$ and $\mathbf{w}_1=(3,2,-4,2)^T, \mathbf{w}_2=(2,3,-7,4)^T, \mathbf{w}_3=(0,3,-3,1)^T$ are two bases for the same three-dimensional subspace $V \subset \mathbb{R}^4$.

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03:19

Problem 30

Define $f(x)= \begin{cases}e^{-1 / x}, & x>0, \\ 0, & x \leq 0 .\end{cases}$
(a) Prove that all derivatives of $f$ vanish at the origin: $f^{(n)}(0)=0$ for $n=0,1,2, \ldots$.
(b) Prove that $f(x)$ is not analytic by showing that its Taylor series at $a=0$ does not converge to $f(x)$ when $x>0$.

Jack Chen
Jack Chen
Numerade Educator
04:45

Problem 30

Prove parts (b) and (c) of Theorem 2.21.

Ahmad Reda
Ahmad Reda
Numerade Educator
06:26

Problem 30

(a) Prove that if $A$ is a symmetric matrix, then $\operatorname{ker} A=\operatorname{coker} A$ and $\operatorname{img} A=\operatorname{coimg} A$. (b) Use this observation to produce bases for the four fundamental subspaces associated with $A=\left(\begin{array}{lll}1 & 2 & 0 \\ 2 & 6 & 2 \\ 0 & 2 & 2\end{array}\right)$.
(c) Is the converse to part (a) true?

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
06:15

Problem 31

Let $f(x)=\frac{1}{1+x^2}$. (a) Find the Taylor series of $f$ at $a=0$. (b) Prove that the Taylor series converges for $|x|<1$, but diverges for $|x| \geq 1$. (c) Prove that $f(x)$ is analytic at $x=0$.

Bobby Barnes
Bobby Barnes
University of North Texas
03:45

Problem 31

(a) Prove that if $\mathbf{v}_1, \ldots, \mathbf{v}_m$ are linearly independent, then every subset, e.g., $\mathbf{v}_1, \ldots, \mathbf{v}_k$ with $k<m$, is also linearly independent. (b) Does the same hold true for linearly dependent vectors? Prove or give a counterexample.

Shu-Ting Huang
Shu-Ting Huang
Numerade Educator
02:34

Problem 31

(a) Write down a matrix of rank $r$ whose first $r$ rows do not form a basis for its row space. (b) Can you find an example that can be reduced to row echelon form without any row interchanges?

Victor Salazar
Victor Salazar
Numerade Educator
01:35

Problem 32

(a) Determine whether the polynomials $f_1(x)=x^2-3, f_2(x)=2-x, f_3(x)=(x-1)^2$, are linearly independent or linearly dependent.
(b) Do they span the vector space of all quadratic polynomials?

WM
William Mead
Numerade Educator
View

Problem 32

Let $A$ be a $4 \times 4$ matrix and let $U$ be its row echelon form. (a) Suppose columns 1,2 , 4 of $U$ form a basis for its image. Do columns 1, 2, 4 of $A$ form a basis for its image? If so, explain why; if not, construct a counterexample. (b) Suppose rows 1, 2, 3 of $U$ form a basis for its coimage. Do rows $1,2,3$ of $A$ form a basis for its coimage? If so, explain why; if not, construct a counterexample. (c) Suppose you find a basis for $\operatorname{ker} U$. Is it also a basis for ker $A$ ? (d) Suppose you find a basis for coker $U$. Is it also a basis for coker $A$ ?

Victor Salazar
Victor Salazar
Numerade Educator
00:54

Problem 33

Determine whether the given functions are linearly independent or linearly dependent:
(a) $2-x^2, 3 x, x^2+x-2$, (b) $3 x-1, x(2 x+1), x(x-1) ;$ (c) $e^x, e^{x+1}$; (d) $\sin x$, $\sin (x+1) ;(e) e^x, e^{x+1}, e^{x+2} ;$ (f) $\sin x, \sin (x+1), \sin (x+2) ;(g) e^x, x e^x, x^2 e^x$;
(h) $e^x, e^{2 x}, e^{3 x}$; (i) $x+y, x-y+1, x+3 y+2$ - these are functions of two variables.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:38

Problem 33

Can you devise a nonzero matrix whose row echelon form is the same as the row echelon form of its transpose?

Ahmad Reda
Ahmad Reda
Numerade Educator
04:11

Problem 34

Show that the functions $f(x)=x$ and $g(x)=|x|$ are linearly independent when considered as functions on all of $\mathbb{R}$, but are linearly dependent when considered as functions defined only on $\mathbb{R}^{+}=\{x>0\}$.

Ibrahima Barry
Ibrahima Barry
Numerade Educator
01:01

Problem 34

Explain why the elementary row operations of types \#2 and \#3 do not change the coimage of a matrix.

Heather Zimmers
Heather Zimmers
Numerade Educator
07:41

Problem 35

(a) Prove that the polynomials $p_i(x)=\sum_{j=0}^n a_{i j} x^j$ for $i=1, \ldots, k$ are linearly independent if and only if the $k \times(n+1)$ matrix $A$ whose entries are their coefficients $a_{i j}, 1 \leq i \leq k, 0 \leq j \leq n$, has rank $k$. (b) Formulate a similar matrix condition for testing whether another polynomial $q(x)$ lies in their span.
(c) Use (a) to determine whether $p_1(x)=x^3-1, p_2(x)=x^3-2 x+4, p_3(x)=x^4-4 x, p_4(x)=x^2+1$, $p_5(x)=-x^4+4 x^3+2 x+1$ are linearly independent or linearly dependent. (d) Does the polynomial $q(x)=x^3$ lie in their span? If so find a linear combination that adds up to $q(x)$.

Donald Albin
Donald Albin
Numerade Educator
02:25

Problem 35

Let $A$ be an $m \times n$ matrix. Prove that $\operatorname{img} A=\mathbb{R}^m$ if and only if $\operatorname{rank} A=m$.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 36

The Fundamental Theorem of Algebra, [26], states that a non-zero polynomial of degree $n$ has at most $n$ distinct real roots, that is, real numbers $x$ such that $p(x)=0$. Use this fact to prove linear independence of the monomial functions $1, x, x^2, \ldots, x^n$.
Remark. An elementary proof of the latter fact can be found in Exercise 5.5.38.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:01

Problem 36

Prove or give a counterexample: If $U$ is the row echelon form of $A$, then $\operatorname{img} U=\operatorname{img} A$.

WM
William Mead
Numerade Educator

Problem 37

(a) Let $x_1, x_2, \ldots, x_n$ be a set of distinct sample points. Prove that the functions $f_1(x), \ldots, f_k(x)$ are linearly independent if their sample vectors $\mathbf{f}_1, \ldots, \mathbf{f}_k$ are linearly independent vectors in $\mathbb{R}^n$. (b) Give an example of linearly independent functions that have linearly dependent sample vectors. (c) Use this method to prove that the functions $1, \cos x$, $\sin x, \cos 2 x, \sin 2 x$, are linearly independent. Hint: You need at least 5 sample points.

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Problem 37

(a) Devise an alternative method for finding a basis of the coimage of a matrix. (b) Use your method to find a basis for the coimage of $\left(\begin{array}{rrrr}1 & 3 & -5 & 2 \\ 2 & -1 & 1 & -4 \\ 4 & 5 & -9 & 2\end{array}\right)$. Is it the same basis as found by the method in the text?

Victor Salazar
Victor Salazar
Numerade Educator
00:56

Problem 38

Suppose $\mathbf{f}_1(t), \ldots, \mathbf{f}_k(t)$ are vector-valued functions from $\mathbb{R}$ to $\mathbb{R}^n$. (a) Prove that if $\mathbf{f}_1\left(t_0\right), \ldots, \mathbf{f}_k\left(t_0\right)$ are linearly independent vectors in $\mathbb{R}^n$ at one point $t_0$, then $\mathbf{f}_1(t), \ldots, \mathbf{f}_k(t)$ are linearly independent functions. (b) Show that $\mathbf{f}_1(t)=\left(\begin{array}{l}1 \\ t\end{array}\right)$ and $\mathbf{f}_2(t)=\left(\begin{array}{c}2 t-1 \\ 2 t^2-t\end{array}\right)$ are linearly independent functions, even though at each $t_0$, the vectors $\mathbf{f}_1\left(t_0\right), \mathbf{f}_2\left(t_0\right)$ are linearly dependent. Therefore, the converse to the result in part (a) is not valid.

Christian Otero
Christian Otero
Numerade Educator
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Problem 38

Prove that $\operatorname{ker} A \subseteq \operatorname{ker} A^2$. More generally, prove $\operatorname{ker} A \subseteq \operatorname{ker} B A$ for every compatible matrix $B$.

Victor Salazar
Victor Salazar
Numerade Educator
02:05

Problem 39

The Wronskian of a pair of differentiable functions $f(x), g(x)$ is the scalar function
$$
W[f(x), g(x)]=\operatorname{det}\left(\begin{array}{rr}
f(x) & g(x) \\
f^{\prime}(x) & g^{\prime}(x)
\end{array}\right)=f(x) g^{\prime}(x)-f^{\prime}(x) g(x) .
$$
(a) Prove that if $f, g$ are linearly dependent, then $W[f(x), g(x)] \equiv 0$. Hence, if $W[f(x), g(x)] \not \equiv 0$, then $f, g$ are linearly independent. (b) Let $f(x)=x^3, g(x)=|x|^3$. Prove that $f, g \in \mathrm{C}^2$ are twice continuously differentiable and linearly independent, but $W[f(x), g(x)] \equiv 0$. Thus, the Wronskian is not a fool-proof test for linear independence.
Remark. It can be proved, [7], that if $f, g$ both saticfy a secend order linear ordinary differential equation, then Page 123 ir 702 everdent @ And +ly if $W[f(x), g(x)] \equiv 0$.

A M
A M
Numerade Educator

Problem 39

Prove that $\operatorname{img} A \supseteq \operatorname{img} A^2$. More generally, prove $\operatorname{img} A \supseteq \operatorname{img}(A B)$ for every compatible matrix $B$.

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Problem 40

Suppose $A$ is an $m \times n$ matrix, and $B$ and $C$ are nonsingular matrices of sizes $m \times m$ and $n \times n$, respectively. Prove that $\operatorname{rank} A=\operatorname{rank} B A=\operatorname{rank} A C=\operatorname{rank} B A C$.

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Problem 41

True or false: If $\operatorname{ker} A=\operatorname{ker} B, \operatorname{then} \operatorname{rank} A=\operatorname{rank} B$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 42

Let $A$ and $B$ be matrices of respective sizes $m \times n$ and $n \times p$.
(a) Prove that $\operatorname{dim} \operatorname{ker}(A B) \leq \operatorname{dim} \operatorname{ker} A+\operatorname{dim} \operatorname{ker} B$.
(b) Prove the Sylvester Inequalities $\operatorname{rank} A+\operatorname{rank} B-n \leq \operatorname{rank}(A B) \leq \min \{\operatorname{rank} A, \operatorname{rank} B\}$.

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Problem 43

Suppose $A$ is a nonsingular $n \times n$ matrix. (a) Prove that every $n \times(n+k)$ matrix of the form ( $A B$ ), where $B$ has size $n \times k$, has rank $n$. (b) Prove that every $(n+k) \times n$ matrix of the form $\left(\begin{array}{l}A \\ C\end{array}\right)$, where $C$ has size $k \times n$, has rank $n$.

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09:00

Problem 44

Let $A$ be an $m \times n$ matrix of rank $r$. Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_n$ are a basis for $\mathbb{R}^n$ such that $\mathbf{v}_{r+1}, \ldots, \mathbf{v}_n$ form a basis for $\operatorname{ker} A$. Prove that $\mathbf{w}_1=A \mathbf{v}_1, \ldots, \mathbf{w}_r=A \mathbf{v}_r$ form a basis for $\operatorname{img} A$.

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 45

(a) Suppose $A, B$ are $m \times n$ matrices such that $\operatorname{ker} A=\operatorname{ker} B$. Prove that there is a nonsingular $m \times m$ matrix $M$ such that $M A=B$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 46

(a) Let $A$ be an $m \times n$ matrix and let $V$ be a subspace of $\mathbb{R}^n$. Show that $W=A V=$ $\{A \mathbf{v} \mid \mathbf{v} \in V\}$ forms a subspace of $\operatorname{img} A$. (b) If $\operatorname{dim} V=k$, show that $\operatorname{dim} W \leq \min \{k, r\}$, where $r=\operatorname{rank} A$. Give an example in which $\operatorname{dim}(A V)<\operatorname{dim} V$.

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01:14

Problem 47

(a) Show that an $m \times n$ matrix has a left inverse if and only if it has rank $n$.

(c) Conclude that only nonsingular square matrices have both left and right inverses.

Victor Salazar
Victor Salazar
Numerade Educator