Contemporary Abstract Algebra

Joseph Gallian

Chapter 19 Vector Spaces - all with Video Answers

Educators

Chapter Questions

Problem 1

Verify that each of the sets in Examples $1-4$ satisfies the axioms for a vector space. Find a basis for each of the vector spaces in Examples $1-4$

Derrick Danso

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02:14

(Subspace Test) Prove that a nonempty subset $U$ of a vector space $V$ over a field $F$ is a subspace of $V$ if, for every $u$ and $u^{\prime}$ in $U$ and every $a$ in $F, u+u^{\prime} \in U$ and $a u \in U$. (In words, a nonempty set $U$ is a subspace of $V$ if it is closed under the two operations of $V .)$

Harshita Goel

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Problem 3

Verify that the set in Example 6 is a subspace. Find a basis for this subspace. Is $\left\{x^{2}+x+1, x+5,3\right\}$ a basis?

Nick Johnson

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06:36

Problem 4

Verify that the set $\left\langle v_{1}, v_{2}, \ldots, v_{n}\right\rangle$ defined in Example 7 is a subspace.

Khoobchandra Agrawal

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01:04

Problem 5

Determine whether or not the set $\{(2,-1,0),(1,2,5),(7,-1,5)\}$ is linearly independent over $\mathbf{R}$.

Shima Shaw

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01:01

Problem 6

Determine whether or not the set
$$
\left\{\left[\begin{array}{ll}
2 & 1 \\
1 & 0
\end{array}\right],\left[\begin{array}{ll}
0 & 1 \\
1 & 2
\end{array}\right],\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\right\}
$$
is linearly independent over $Z_{5}$.

Shima Shaw

Numerade Educator

01:01

Problem 7

If $\{u, v, w\}$ is a linearly independent subset of a vector space, show that $\{u, u+v, u+v+w\}$ is also linearly independent.

Shima Shaw

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01:47

Problem 8

If $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ is a linearly dependent set of vectors, prove that one of these vectors is a linear combination of the other.

Manisha Sarker

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04:15

Problem 9

(Every spanning collection contains a basis.) If $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ spans a vector space $V$, prove that some subset of the $v$ 's is a basis for $V$.

Priyanka Sadarangani

Numerade Educator

01:05

Problem 10

(Every independent set is contained in a basis.) Let $V$ be a finitedimensional vector space and let $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ be a linearly independent subset of $V$. Show that there are vectors $w_{1}, w_{2}, \ldots, w_{m}$ such that $\left\{v_{1}, v_{2}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\right\}$ is a basis for $V$.

Hoan Nguyen

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02:49

Problem 11

If $V$ is a vector space over $F$ of dimension 5 and $U$ and $W$ are subspaces of $V$ of dimension 3, prove that $U \cap W \neq\{0\}$. Generalize.

Nick Johnson

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Problem 12

Show that the solution set to a system of equations of the form where the $a$ 's are real, is a subspace of $\mathbf{R}^{n}$.

Victor Salazar

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03:28

Problem 13

Let $V$ be the set of all polynomials over $Q$ of degree 2 together with the zero polynomial. Is $V$ a vector space over $Q$ ?

Mengchun Cai

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02:40

Problem 14

Let $V=\mathbf{R}^{3}$ and $W=\left\{(a, b, c) \in V \mid a^{2}+b^{2}=c^{2}\right\} .$ Is $W$ a subspace of $V ?$ If so, what is its dimension?

Jimmy Yao

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05:30

Problem 15

Let $V=\mathbf{R}^{3}$ and $W=\{(a, b, c) \in V \mid a+b=c\} .$ Is $W$ a subspace of $V ?$ If so, what is its dimension?

Aidan Mcnabb

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05:30

Problem 16

Let $V=\left\{\left[\begin{array}{ll}a & b \\ b & c\end{array}\right] \mid a, b, c \in Q\right\}$. Prove that $V$ is a vector space over $Q$, and find a basis for $V$ over $Q$.

Anthony Ramos

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11:51

Problem 17

Verify that the set $V$ in Example 9 is a vector space over $\mathbf{R}$.

Chris Trentman

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02:31

Problem 18

Let $P=\{(a, b, c) \mid a, b, c \in \mathbf{R}, a=2 b+3 c\}$. Prove that $P$ is a subspace of $\mathbf{R}^{3}$. Find a basis for $P$. Give a geometric description of $P$.

Victor Salazar

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04:15

Problem 19

Let $B$ be a subset of a vector space $V$. Show that $B$ is a basis for $V$ if and only if every member of $V$ is a unique linear combination of the elements of $B$. (This exercise is referred to in this chapter and in Chapter 20.)

Priyanka Sadarangani

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01:25

Problem 20

If $U$ is a proper subspace of a finite-dimensional vector space $V$, show that the dimension of $U$ is less than the dimension of $V$.

Ed Tam

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07:25

Problem 21

Referring to the proof of Theorem $19.1$, prove that $\left\{w_{1}, u_{2}, \ldots, u_{m}\right\}$ spans $V$.

Chris Trentman

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Problem 22

If $V$ is a vector space of dimension $n$ over the field $Z_{p}$, how many elements are in $V ?$

Nick Johnson

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04:12

Problem 23

Let $S=\{(a, b, c, d) \mid a, b, c, d \in \mathbf{R}, a=c, d=a+b\} .$ Find a basis
for $S$.

Gopesh Vishwakarma

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11:42

Problem 24

Let $U$ and $W$ be subspaces of a vector space $V$. Show that $U \cap W$ is a subspace of $V$ and that $U+W=\{u+w \mid u \in U, w \in W\}$ is a subspace of $V$.

E R

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04:15

Problem 25

If a vector space has one basis that contains infinitely many elements, prove that every basis contains infinitely many elements. (This exercise is referred to in this chapter.)

Priyanka Sadarangani

Numerade Educator

16:05

Problem 26

Let $u=(2,3,1), v=(1,3,0)$, and $w=(2,-3,3)$. Since $(1 / 2) u-$ $(2 / 3) v-(1 / 6) w=(0,0,0)$, can we conclude that the set $\{u, v, w\}$ is linearly dependent over $Z_{7}$ ?

Donald Albin

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03:58

Problem 27

Define the vector space analog of group homomorphism and ring homomorphism. Such a mapping is called a linear transformation. Define the vector space analog of group isomorphism and ring isomorphism.

Anthony Ramos

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06:35

Problem 28

Let $T$ be a linear transformation from $V$ to $W$. Prove that the image of $V$ under $T$ is a subspace of $W$.

Michael Jacobsen

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01:13

Problem 29

Let $T$ be a linear transformation of a vector space $V$. Prove that $\{v \in V \mid T(v)=0\}$, the kernel of $T$, is a subspace of $V$.

Arun Bana

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03:47

Problem 30

Let $T$ be a linear transformation of $V$ onto $W$. If $\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}$ spans $V$, show that $\left\{T\left(v_{1}\right), T\left(v_{2}\right), \ldots, T\left(v_{n}\right)\right\}$ spans $W$.

Henry Carnick

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03:58

Problem 31

If $V$ is a vector space over $F$ of dimension $n$, prove that $V$ is isomorphic as a vector space to $F^{n}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \mid a_{i} \in F\right\} .$ (This exercise is referred to in this chapter.)

Anthony Ramos

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02:43

Problem 32

Show that it is impossible to find a basis for the vector space of $n \times n(n>1)$ matrices such that each pair of elements in the basis commutes under multiplication.

Ed Tam

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01:10

Problem 33

Let $P_{n}=\left\{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \mid\right.$ each $a_{i}$ is a real
number $\} .$ Is it possible to have a basis for $P_{n}$ such that every element of the basis has $x$ as a factor?

Keshav Singh

Numerade Educator

01:34

Problem 34

Find a basis for the vector space $\left\{f \in P_{3} \mid f(0)=0\right\}$. (See Exercise 33 for notation.)

Victor Salazar

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04:00

Problem 35

Given that $f$ is a polynomial of degree $n$ in $P_{n}$, show that $\left\{f, f^{\prime}\right.$, $\left.f^{\prime \prime}, \ldots, f^{(n)}\right\}$ is a basis for $P_{n} .\left(f^{(k)}\right.$ denotes the $k$ th derivative of $\left.f .\right)$

Clarissa Noh

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02:04

Problem 36

Prove that for a vector space $V$ over a field that does not have characteristic 2, the hypothesis that $V$ is commutative under addition is redundant.

Viswesh Uppalapati

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01:25

Problem 37

Let $V$ be a vector space over an infinite field. Prove that $V$ is not the union of finitely many proper subspaces of $V$.

Ed Tam

Numerade Educator