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Linear Algebra: A Modern Introduction

David Poole

Chapter 1

Vectors - all with Video Answers

Educators

Section 1

The Geometry and Algebra of Vectors

01:32

Problem 1

Draw the following vectors in standard position in $\mathbb{R}^{2}$ :
(a) $a=\left[\begin{array}{l}3 \\ 0\end{array}\right]$
(b) $\mathbf{b}=\left[\begin{array}{l}2 \\ 3\end{array}\right]$
(c) $c=\left[\begin{array}{r}-2 \\ 3\end{array}\right]$
(d) $\mathbf{d}=\left[\begin{array}{r}3 \\ -2\end{array}\right]$

Liuxi Sun
Liuxi Sun
Numerade Educator
00:36

Problem 2

Draw the vectors in Exercise 1 with their tails at the point (2,-3)

Ashley Hanson
Ashley Hanson
Numerade Educator
01:10

Problem 3

Draw the following vectors in standard position in $\mathbb{R}^{3}$ :
(a) $a=[0,2,0]$
(b) $\mathbf{b}=[3,2,1]$
(c) $\mathbf{c}=[1,-2,1]$
(d) $\mathbf{d}=[-1,-1,-2]$

Breanna Ollech
Breanna Ollech
Numerade Educator
02:37

Problem 4

If the vectors in Exercise 3 are translated so that their heads are at the point $(3,2,1),$ find the points that correspond to their tails.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:26

Problem 5

For each of the following pairs of points, draw the vector $\overrightarrow{A B}$. Then compute and redraw $\overrightarrow{A B}$ as a vector in standard position.
(a) $A=(1,-1), B=(4,2)$
(b) $A=(0,-2), B=(2,-1)$
(c) $A=\left(2, \frac{3}{2}\right), B=\left(\frac{1}{2}, 3\right)$
(d) $A=\left(\begin{array}{l}1 \\ 3\end{array}, \frac{1}{3}\right), B=\left(\frac{1}{6}, \frac{1}{2}\right)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:18

Problem 6

A hiker walks $4 \mathrm{km}$ north and then $5 \mathrm{km}$ northeast. Draw displacement vectors representing the hiker's trip and draw a vector that represents the hiker's net displacement from the starting point.

Narayan Hari
Narayan Hari
Numerade Educator
04:24

Problem 7

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$a+b$$

Emily Anderson
Emily Anderson
Numerade Educator
04:24

Problem 8

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$b-c$$

Emily Anderson
Emily Anderson
Numerade Educator
04:24

Problem 9

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$\mathrm{d}-\mathrm{c}$$

Emily Anderson
Emily Anderson
Numerade Educator
04:24

Problem 10

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$a+d$$

Emily Anderson
Emily Anderson
Numerade Educator
03:33

Problem 11

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$2 a+3 c$$

Manisha Sarker
Manisha Sarker
Numerade Educator
00:45

Problem 12

Refer to the vectors in Exercise $1 .$ Compute the indicated vectors and also show how the results can be obtained geometrically.
$$3 b-2 c+d$$

Amy Jiang
Amy Jiang
Numerade Educator
04:18

Problem 13

Find the components of the vectors $\mathbf{u}, \mathbf{v}, \mathbf{u}+\mathbf{v},$ and $\mathbf{u}-\mathbf{v},$ where $\mathbf{u}$ and $\mathbf{v}$ are as shown in Figure 1.23

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:45

Problem 14

In Figure $1.24, A, B, C, D, E,$ and $F$ are the vertices of a regular hexagon centered at the origin. Express each of the following vectors in terms of $\mathbf{a}=\overrightarrow{O A}$ and $\mathbf{b}=\overrightarrow{O B}$
(a) $\overrightarrow{A B}$
(b) $\overrightarrow{B C}$
(c) $\overrightarrow{A D}$
(d) $\overrightarrow{C F}$
(e) $\overrightarrow{A C}$
(f) $\overrightarrow{B C}+\overrightarrow{D E}+\overrightarrow{F A}$

Akshaya Rs
Akshaya Rs
Numerade Educator
01:04

Problem 15

Simplify the given vector expression. Indicate which properties in Theorem 1.1 you use.
$$2(a-3 b)+3(2 b+a)$$

Vysakh M
Vysakh M
Numerade Educator
01:57

Problem 16

Simplify the given vector expression. Indicate which properties in Theorem 1.1 you use.
$$-3(a-c)+2(a+2 b)+3(c-b)$$

Vysakh M
Vysakh M
Numerade Educator
01:32

Problem 17

Solve for the vector $\mathbf{x}$ in terms of the vectors a and $\mathbf{b}$.
$$\mathbf{x}-\mathbf{a}=2(\mathbf{x}-2 \mathbf{a})$$

Vysakh M
Vysakh M
Numerade Educator
01:32

Problem 18

Solve for the vector $\mathbf{x}$ in terms of the vectors a and $\mathbf{b}$.
$$x+2 a-b=3(x+a)-2(2 a-b)$$

Vysakh M
Vysakh M
Numerade Educator
01:12

Problem 19

Draw the coordinate axes relative to u and v and locate w.
$$\mathbf{u}=\left[\begin{array}{r}
1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \mathbf{w}=2 \mathbf{u}+3 \mathbf{v}$$

AG
Ankit Gupta
Numerade Educator
01:22

Problem 20

Draw the coordinate axes relative to u and v and locate w.
$$\mathbf{u}=\left[\begin{array}{r}
-2 \\
1
\end{array}\right], \mathbf{v}=\left[\begin{array}{r}
2 \\
-2
\end{array}\right], \mathbf{w}=-\mathbf{u}-2 \mathbf{v}$$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
00:32

Problem 21

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find $\mathbf{w}$ as a linear combination of u and v.
$$\mathbf{u}=\left[\begin{array}{r}
1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \mathbf{w}=\left[\begin{array}{l}
2 \\
6
\end{array}\right]$$

Ashley Hanson
Ashley Hanson
Numerade Educator
00:32

Problem 22

Draw the standard coordinate axes on the same diagram as the axes relative to u and v. Use these to find $\mathbf{w}$ as a linear combination of u and v.
$$\mathbf{u}=\left[\begin{array}{r}
-2 \\
3
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
2 \\
1
\end{array}\right], \mathbf{w}=\left[\begin{array}{l}
2 \\
9
\end{array}\right]$$

Ashley Hanson
Ashley Hanson
Numerade Educator
00:38

Problem 23

Draw diagrams to illustrate properties (d) and (e) of Theorem 1.1

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:44

Problem 24

Give algebraic proofs of properties (d) through (g) of Theorem 1.1

Linh Vu
Linh Vu
Numerade Educator
01:27

Problem 25

u and v are binary vectors. Find $\mathbf{u}+\mathbf{v}$ in each case.
$$\mathbf{u}=\left[\begin{array}{l}
0 \\
1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1
\end{array}\right]$$

Thane Stiles
Thane Stiles
Numerade Educator
04:19

Problem 26

u and v are binary vectors. Find $\mathbf{u}+\mathbf{v}$ in each case.
$$\mathbf{u}=\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:27

Problem 27

u and v are binary vectors. Find $\mathbf{u}+\mathbf{v}$ in each case.
$$\mathbf{u}=[1,0,1,1], \mathbf{v}=[1,1,1,1]$$

Thane Stiles
Thane Stiles
Numerade Educator
00:55

Problem 28

u and v are binary vectors. Find $\mathbf{u}+\mathbf{v}$ in each case.
$$\mathbf{u}=[1,1,0,1,0], \mathbf{v}=[0,1,1,1,0]$$

Subhakanta Sahoo
Subhakanta Sahoo
Numerade Educator
03:27

Problem 29

Write out the addition and multiplication tables for $\mathbb{Z}_{4}$.

Ruby P
Ruby P
Numerade Educator
02:00

Problem 30

Write out the addition and multiplication tables for $\mathbb{Z}_{5}$.

James Chok
James Chok
Numerade Educator
00:41

Problem 31

Perform the indicated calculations.
$$-2+2+2 \text { in } \mathbb{Z}_{3}$$

Nick Johnson
Nick Johnson
Numerade Educator
00:14

Problem 32

Perform the indicated calculations.
$$2 \cdot 2 \cdot 2 \text { in } \mathbb{Z}_{3}$$.

Amy Jiang
Amy Jiang
Numerade Educator
00:29

Problem 33

Perform the indicated calculations.
$$2(2+1+2) \text { in } \mathbb{Z}_{3}$$.

Amy Jiang
Amy Jiang
Numerade Educator
00:24

Problem 34

Perform the indicated calculations.
$$3+1+2+3 \text { in } \mathbb{Z}_{4}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:19

Problem 35

Perform the indicated calculations.
$$2 \cdot 3 \cdot 2 \text { in } \mathbb{Z}_{4}$$

Matt Gibson
Matt Gibson
Numerade Educator
00:14

Problem 36

Perform the indicated calculations.
$$3(3+3+2) \text { in } \mathbb{Z}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:35

Problem 37

Perform the indicated calculations.
$$2+1+2+2+1 \text { in } \mathbb{Z}_{3}, Z_{4}, \text { and } \mathbb{Z}_{5}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:12

Problem 38

Perform the indicated calculations.
$$(3+4)(3+2+4+2) \text { in } \mathbb{Z}_{5}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:52

Problem 39

Perform the indicated calculations.
$$8(6+4+3) \text { in } \mathbb{Z}$$

AG
Ankit Gupta
Numerade Educator
00:16

Problem 40

Perform the indicated calculations.
$$2^{100} \text { in } \mathbb{Z}_{11}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:14

Problem 41

Perform the indicated calculations.
$$[2,1,2]+[2,0,1] \text { in } \mathbb{Z}_{3}^{3}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:11

Problem 42

Perform the indicated calculations.
$$2[2,2,1] \text { in } \mathbb{Z}_{3}^{3}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:39

Problem 43

Perform the indicated calculations.
$$2([3,1,1,2]+[3,3,2,1]) \text { in } \mathbb{Z}_{4}^{4} \text { and } \mathbb{Z}_{5}^{4}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:49

Problem 44

Solve the given equation or indicate that there is no solution.
$$x+3=2 \text { in } \mathbb{Z}_{5}$$

William Greiner
William Greiner
Numerade Educator
00:36

Problem 45

Solve the given equation or indicate that there is no solution.
$$x+5=1 \text { in } \mathbb{Z}_{6}$$

Khanh Ha
Khanh Ha
Numerade Educator
01:23

Problem 46

Solve the given equation or indicate that there is no solution.
$$2 x=1 \text { in } \mathbb{Z}_{3}$$

Andrew John De Los Santos
Andrew John De Los Santos
Numerade Educator
00:31

Problem 47

Solve the given equation or indicate that there is no solution.
$$2 x=1 \text { in } \mathbb{Z}_{4}$$

Amy Jiang
Amy Jiang
Numerade Educator
01:03

Problem 48

Solve the given equation or indicate that there is no solution.
$$2 x=1 \text { in } \mathbb{Z}_{5}$$

William Greiner
William Greiner
Numerade Educator
01:48

Problem 49

Solve the given equation or indicate that there is no solution.
$$3 x=4 \text { in } \mathbb{Z}_{5}$$

Edward Downes
Edward Downes
Numerade Educator
00:58

Problem 50

Solve the given equation or indicate that there is no solution.
$$3 x=4 \text { in } \mathbb{Z}_{6}$$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:26

Problem 51

Solve the given equation or indicate that there is no solution.
$$6 x=5 \text { in } \mathbb{Z}_{8}$$

Angela Guo
Angela Guo
Numerade Educator
01:48

Problem 52

Solve the given equation or indicate that there is no solution.
$$8 x=9 \text { in } \mathbb{Z}_{11}$$

Edward Downes
Edward Downes
Numerade Educator
00:49

Problem 53

Solve the given equation or indicate that there is no solution.
$$2 x+3=2 \text { in } \mathbb{Z}_{5}$$

William Greiner
William Greiner
Numerade Educator
00:57

Problem 54

Solve the given equation or indicate that there is no solution.
$$4 x+5=2 \text { in } \mathbb{Z}_{6}$$

Brandon Fox
Brandon Fox
Numerade Educator
01:18

Problem 55

Solve the given equation or indicate that there is no solution.
$$6 x+3=1 \text { in } \mathbb{Z}_{8}$$

Katelyn Vandeaver
Katelyn Vandeaver
Numerade Educator
01:45

Problem 56

(a) For which values of $a$ does $x+a=0$ have a solution in $\mathbb{Z}_{5}$ ?
(b) For which values of $a$ and $b$ does $x+a=b$ have a solution in $\mathbb{Z}_{6} ?$
(c) For which values of $a, b,$ and $m$ does $x+a=b$ have a solution in $\mathbb{Z}_{m}$ ?

AG
Ankit Gupta
Numerade Educator
00:51

Problem 57

(a) For which values of $a$ does $a x=1$ have a solution in $\mathbb{Z}_{5} ?$
(b) For which values of $a$ does $a x=1$ have a solution in $\mathbb{Z}_{6} ?$
(c) For which values of $a$ and $m$ does $a x=1$ have a solution in $\mathbb{Z}_{m} ?$

Emily Schilz
Emily Schilz
Numerade Educator