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Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 6

Orthogonality and Least Square - all with Video Answers

Educators

GM

Section 1

Inner Product, Length, and Orthogonality

02:45

Problem 1

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\mathbf{u} \cdot \mathbf{u}, \mathbf{v} \cdot \mathbf{u},$ and $\frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}}$

Ryan Sander
Ryan Sander
Numerade Educator
01:34

Problem 2

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\mathbf{w} \cdot \mathbf{w}, \mathbf{x} \cdot \mathbf{w},$ and $\frac{\mathbf{X} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}$

Ashley Boni
Ashley Boni
Numerade Educator
03:03

Problem 3

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\frac{1}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}$

Ryan Sander
Ryan Sander
Numerade Educator
00:59

Problem 4

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\frac{1}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$

Ashley Boni
Ashley Boni
Numerade Educator
03:43

Problem 5

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}$

Ryan Sander
Ryan Sander
Numerade Educator
01:47

Problem 6

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\left(\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{x} \cdot \mathbf{x}}\right) \mathbf{x}$

Ashley Boni
Ashley Boni
Numerade Educator
00:51

Problem 7

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\|\mathbf{w}\|$

Ryan Sander
Ryan Sander
Numerade Educator
00:47

Problem 8

Compute the quantities in Exercises $1-8$ using the vectors
$$
\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]
$$
$\|\mathbf{x}\|$

Ashley Boni
Ashley Boni
Numerade Educator
03:50

Problem 9

In Exercises $9-12,$ find a unit vector in the direction of the given vector.
$$
\left[\begin{array}{r}{-30} \\ {40}\end{array}\right]
$$

Ryan Sander
Ryan Sander
Numerade Educator
01:36

Problem 10

In Exercises $9-12,$ find a unit vector in the direction of the given vector.
$$
\left[\begin{array}{r}{-6} \\ {4} \\ {-3}\end{array}\right]
$$

Ashley Boni
Ashley Boni
Numerade Educator
04:47

Problem 11

In Exercises $9-12,$ find a unit vector in the direction of the given vector.
$$
\left[\begin{array}{c}{7 / 4} \\ {1 / 2} \\ {1}\end{array}\right]
$$

Ryan Sander
Ryan Sander
Numerade Educator
02:03

Problem 12

In Exercises $9-12,$ find a unit vector in the direction of the given vector.
$$
\left[\begin{array}{c}{8 / 3} \\ {2}\end{array}\right]
$$

Ashley Boni
Ashley Boni
Numerade Educator
04:18

Problem 13

Find the distance between $\mathbf{x}=\left[\begin{array}{c}{10} \\ {-3}\end{array}\right]$ and $\mathbf{y}=\left[\begin{array}{c}{-1} \\ {-5}\end{array}\right]$

Ryan Sander
Ryan Sander
Numerade Educator
01:30

Problem 14

Find the distance between $\mathbf{u}=\left[\begin{array}{r}{0} \\ {-5} \\ {2}\end{array}\right]$ and $\mathbf{z}=\left[\begin{array}{r}{-4} \\ {-1} \\ {8}\end{array}\right]$

Ashley Boni
Ashley Boni
Numerade Educator
01:39

Problem 15

Determine which pairs of vectors in Exercises $15-18$ are orthogonal.
$$
\mathbf{a}=\left[\begin{array}{r}{8} \\ {-5}\end{array}\right], \mathbf{b}=\left[\begin{array}{l}{-2} \\ {-3}\end{array}\right]
$$

Kenwa Nandi
Kenwa Nandi
Numerade Educator
00:48

Problem 16

Determine which pairs of vectors in Exercises $15-18$ are orthogonal.
$$
\mathbf{u}=\left[\begin{array}{r}{12} \\ {3} \\ {-5}\end{array}\right], \mathbf{v}=\left[\begin{array}{r}{2} \\ {-3} \\ {3}\end{array}\right]
$$

Ashley Boni
Ashley Boni
Numerade Educator
02:07

Problem 17

Determine which pairs of vectors in Exercises $15-18$ are orthogonal.
$$
\mathbf{u}=\left[\begin{array}{r}{3} \\ {2} \\ {-5} \\ {0}\end{array}\right], \mathbf{v}=\left[\begin{array}{r}{-4} \\ {1} \\ {-2} \\ {6}\end{array}\right]
$$

Kenwa Nandi
Kenwa Nandi
Numerade Educator
01:04

Problem 18

Determine which pairs of vectors in Exercises $15-18$ are orthogonal.
$$
\mathbf{y}=\left[\begin{array}{r}{-3} \\ {7} \\ {4} \\ {0}\end{array}\right], \mathbf{z}=\left[\begin{array}{r}{1} \\ {-8} \\ {15} \\ {-7}\end{array}\right]
$$

Ashley Boni
Ashley Boni
Numerade Educator
09:15

Problem 19

In Exercises 19 and $20,$ all vectors are in $\mathbb{R}^{n} .$ Mark each statement True or False. Justify each answer.
a. $\mathbf{v} \cdot \mathbf{v}=\|\mathbf{v}\|^{2}$
b. For any scalar $c, \mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v})$
c. If the distance from $\mathbf{u}$ to $\mathbf{v}$ equals the distance from $\mathbf{u}$ to $\quad-\mathbf{v},$ then $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
d. For a square matrix $A,$ vectors in Col $A$ are orthogonal to vectors in Nul $A .$
e. If vectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}$ span a subspace $W$ and if $\mathbf{x}$ is
orthogonal to each $\mathbf{v}_{j}$ for $j=1, \ldots, p,$ then $\mathbf{x}$ is in $W^{\perp}$ .

Kenwa Nandi
Kenwa Nandi
Numerade Educator
05:09

Problem 20

In Exercises 19 and $20,$ all vectors are in $\mathbb{R}^{n} .$ Mark each statement True or False. Justify each answer.
a. $\mathbf{u} \cdot \mathbf{v}-\mathbf{v} \cdot \mathbf{u}=0$
b. For any scalar $c,\|c \mathbf{v}\|=c\|\mathbf{v}\|$
c. If $\mathbf{x}$ is orthogonal to every vector in a subspace $W,$ then $\mathbf{x}$ $\quad$ is in $W^{\perp}$
d. If $\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}=\|\mathbf{u}+\mathbf{v}\|^{2}$ , then $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
e. For an $m \times n$ matrix $A,$ vectors in the null space of $A$ are orthogonal to vectors in the row space of $A .$

Ashley Boni
Ashley Boni
Numerade Educator
05:22

Problem 21

Use the transpose definition of the inner product to verify parts $(b)$ and $(c)$ of Theorem $1 .$ Mention the appropriate facts from Chapter 2 .

Ahmad Reda
Ahmad Reda
Numerade Educator
01:17

Problem 22

Let $\mathbf{u}=\left(u_{1}, u_{2}, u_{3}\right) .$ Explain why $\mathbf{u} \cdot \mathbf{u} \geq 0 .$ When is $\mathbf{u} \cdot \mathbf{u}=0 ?$

Ashley Boni
Ashley Boni
Numerade Educator
04:50

Problem 23

Let $\mathbf{u}=\left[\begin{array}{r}{2} \\ {-5} \\ {-1}\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{r}{-7} \\ {-4} \\ {6}\end{array}\right] .$ Compute and compare $\mathbf{u} \cdot \mathbf{v},\|\mathbf{u}\|^{2},\|\mathbf{v}\|^{2},$ and $\|\mathbf{u}+\mathbf{v}\|^{2} .$ Do not use the Pythagorean
Theorem.

Ruby P
Ruby P
Numerade Educator
02:19

Problem 24

Verify the parallelogram law for vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{n} :$
$\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}$

Ashley Boni
Ashley Boni
Numerade Educator
02:32

Problem 25

Let $\mathbf{v}=\left[\begin{array}{l}{a} \\ {b}\end{array}\right] .$ Describe the set $H$ of vectors $\left[\begin{array}{l}{x} \\ {y}\end{array}\right]$ that are orthogonal to $\mathbf{v} .[\text { Hint: Consider } \mathbf{v}=\mathbf{0} \text { and } \mathbf{v} \neq \mathbf{0} .]$

Kenwa Nandi
Kenwa Nandi
Numerade Educator
02:23

Problem 26

Let $\mathbf{u}=\left[\begin{array}{r}{5} \\ {-6} \\ {7}\end{array}\right],$ and let $W$ be the set of all $\mathbf{x}$ in $\mathbb{R}^{3}$ such that $\mathbf{u} \cdot \mathbf{x}=0 .$ What theorem in Chapter 4 can be used to show that $W$ is a subspace of $\mathbb{R}^{3} ?$ Describe $W$ in geometric language.

Ashley Boni
Ashley Boni
Numerade Educator
02:25

Problem 27

Suppose a vector $\mathbf{y}$ is orthogonal to vectors $\mathbf{u}$ and $\mathbf{v} .$ Show
that $\mathbf{y}$ is orthogonal to the vector $\mathbf{u}+\mathbf{v} .$

Kenwa Nandi
Kenwa Nandi
Numerade Educator
02:12

Problem 28

Suppose $\mathbf{y}$ is orthogonal to $\mathbf{u}$ and $\mathbf{v} .$ Show that $\mathbf{y}$ is orthogonal to every $\mathbf{w}$ in Span $\{\mathbf{u}, \mathbf{v}\} .[\text { Hint: An arbitrary } \mathbf{w} \text { in } \mathrm{Span}\{\mathbf{u}, \mathbf{v}\}$ has the form $\mathbf{w}=c_{1} \mathbf{u}+c_{2} \mathbf{v} .$ Show that $\mathbf{y}$ is orthogonal to such a vector $\mathbf{w} . ]$

Ashley Boni
Ashley Boni
Numerade Educator
03:54

Problem 29

Let $W=\operatorname{Span}\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\} .$ Show that if $\mathbf{x}$ is orthogonal to each $\mathbf{v}_{j},$ for $1 \leq j \leq p,$ then $\mathbf{x}$ is orthogonal to every vector in $W .$

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 30

Let $W$ be a subspace of $\mathbb{R}^{n},$ and let $W^{\perp}$ be the set of all vectors orthogonal to $W .$ Show that $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ using the following steps.
a. Take $\mathbf{z}$ in $W^{\perp}$ , and let u represent any element of $W .$ Then $\mathbf{z} \cdot \mathbf{u}=0 .$ Take any scalar $c$ and show that $c \mathbf{z}$ is orthogonal to $\mathbf{u} .$ (Since $\mathbf{u}$ was an arbitrary element of $W,$ this will show that $c \mathbf{z}$ is in $W^{\perp} . )$
b. Take $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ in $W^{\perp},$ and let $\mathbf{u}$ be any element of
$W .$ Show that $\mathbf{z}_{1}+\mathbf{z}_{2}$ is orthogonal to $\mathbf{u} .$ What can you conclude about $\mathbf{z}_{1}+\mathbf{z}_{2} ?$ Why?
c. Finish the proof that $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ .

Victor Salazar
Victor Salazar
Numerade Educator
02:43

Problem 31

Show that if $x$ is in both $W$ and $W^{\perp},$ then $x=0$

Lewis Groves
Lewis Groves
Numerade Educator
01:26

Problem 32

$[\mathbf{M}]$ Construct a pair $\mathbf{u}, \mathbf{v}$ of random vectors in $\mathbb{R}^{4},$ and let
$$
A=\left[\begin{array}{rrrr}{.5} & {.5} & {.5} & {.5} \\ {.5} & {.5} & {-.5} & {-.5} \\ {.5} & {-.5} & {-5} & {-.5} \\ {.5} & {-.5} & {-5} & {.5}\end{array}\right]
$$
a. Denote the columns of $A$ by $a_{1}, \ldots, a_{4} .$ Compute the length of each column, and compute $a_{1} \cdot a_{2},$ $\mathbf{a}_{1} \cdot \mathbf{a}_{3}, \mathbf{a}_{1} \cdot \mathbf{a}_{4}, \mathbf{a}_{2} \cdot \mathbf{a}_{3}, \mathbf{a}_{2} \cdot \mathbf{a}_{4},$ and $\mathbf{a}_{3} \cdot \mathbf{a}_{4}$
b. Compute and compare the lengths of $\mathbf{u}, A \mathbf{u}, \mathbf{v},$ and $A \mathbf{v}$
c. Use equation $(2)$ in this section to compute the cosine of the angle between $\mathbf{u}$ and $\mathbf{v} .$ Compare this with the cosine of the angle between $A \mathbf{u}$ and $A \mathbf{v} .$
d. Repeat parts $(\mathrm{b})$ and (c) for two other pairs of random vectors. What do you conjecture about the effect of $A$ on vectors?

Jie Min
Jie Min
Numerade Educator
01:38

Problem 33

[M] Generate random vectors $\mathbf{x}, \mathbf{y},$ and $\mathbf{v}$ in $\mathbb{R}^{4}$ with integer entries (and $\mathbf{v} \neq \mathbf{0} ),$ and compute the quantities
$
\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v},\left(\frac{\mathbf{y} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}, \frac{(\mathbf{x}+\mathbf{y}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}, \frac{(10 \mathbf{x}) \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}
$
Repeat the computations with new random vectors $\mathbf{x}$ and $\mathbf{y} .$ What do you conjecture about the mapping $\mathbf{x} \mapsto T(\mathbf{x})=$ $\left(\frac{\mathbf{x} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}(\text { for } \mathbf{v} \neq \mathbf{0}) ?$ Verify your conjecture algebraically.

Manik Pulyani
Manik Pulyani
Numerade Educator
05:21

Problem 34

[M] Let $A=\left[\begin{array}{rrrrr}{-6} & {3} & {-27} & {-33} & {-13} \\ {6} & {-5} & {25} & {28} & {14} \\ {8} & {-6} & {34} & {38} & {18} \\ {12} & {-10} & {50} & {41} & {23} \\ {14} & {-21} & {49} & {29} & {33}\end{array}\right]$ construct a matrix $R$ whose rows form a basis for Row $A$ (see
Section 4.6 for details). Perform a matrix computation with $N$ and $R$ that illustrates a fact from Theorem 3 .

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator