• Home
  • Textbooks
  • Linear Algebra and Its Applications
  • The Geometry of Vector Spaces

Linear Algebra and Its Applications

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

Chapter 8

The Geometry of Vector Spaces - all with Video Answers

Educators

Section 1

Affine Combinations

04:58

Problem 1

In Exercises $1-4,$ write $\mathbf{y}$ as an affine combination of the other points listed, if possible.
$$
\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {2}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{-2} \\ {2}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{l}{0} \\ {4}\end{array}\right], \mathbf{v}_{4}=\left[\begin{array}{l}{3} \\ {7}\end{array}\right], \mathbf{y}=\left[\begin{array}{l}{5} \\ {3}\end{array}\right]
$$

Michael Jacobsen
Michael Jacobsen
Numerade Educator
02:32

Problem 2

In Exercises $1-4,$ write $\mathbf{y}$ as an affine combination of the other points listed, if possible.
$$
\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {1}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{l}{3} \\ {2}\end{array}\right], \mathbf{y}=\left[\begin{array}{l}{5} \\ {7}\end{array}\right]
$$

Gideon Idumah
Gideon Idumah
Numerade Educator
04:23

Problem 3

In Exercises $1-4,$ write $\mathbf{y}$ as an affine combination of the other points listed, if possible.
$$
\mathbf{v}_{1}=\left[\begin{array}{r}{-3} \\ {1} \\ {1}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{0} \\ {4} \\ {-2}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{4} \\ {-2} \\ {6}\end{array}\right], \mathbf{y}=\left[\begin{array}{r}{17} \\ {1} \\ {5}\end{array}\right]
$$

Michael Jacobsen
Michael Jacobsen
Numerade Educator
03:14

Problem 4

In Exercises $1-4,$ write $\mathbf{y}$ as an affine combination of the other points listed, if possible.
$$
\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{2} \\ {-6} \\ {7}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{l}{4} \\ {3} \\ {1}\end{array}\right], \mathbf{y}=\left[\begin{array}{r}{-3} \\ {4} \\ {-4}\end{array}\right]
$$

Gideon Idumah
Gideon Idumah
Numerade Educator
06:51

Problem 5

In Exercises 5 and $6,$ let $\mathbf{b}_{1}=\left[\begin{array}{l}{2} \\ {1} \\ {1}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{1} \\ {0} \\ {-2}\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}{2} \\ {-5} \\ {1}\end{array}\right]$ and $S=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\} .$ Note that $S$ is an orthogonal basis for $\mathbb{R}^{3} .$ Write each of the given points as an affine combination of the points in the set $S,$ if possible.Hint: Use Theorem 5 in Section 6.2 instead of row reduction to find the weights.
$$
\text { a. }\mathbf{p}_{1}=\left[\begin{array}{l}{3} \\ {8} \\ {4}\end{array}\right] \quad \text { b. } \mathbf{p}_{2}=\left[\begin{array}{r}{6} \\ {-3} \\ {3}\end{array}\right] \quad \text { c. } \mathbf{p}_{3}=\left[\begin{array}{r}{0} \\ {-1} \\ {-5}\end{array}\right]
$$

Michael Jacobsen
Michael Jacobsen
Numerade Educator
06:45

Problem 6

In Exercises 5 and $6,$ let $\mathbf{b}_{1}=\left[\begin{array}{l}{2} \\ {1} \\ {1}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{1} \\ {0} \\ {-2}\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}{2} \\ {-5} \\ {1}\end{array}\right]$ and $S=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\} .$ Note that $S$ is an orthogonal basis for $\mathbb{R}^{3} .$ Write each of the given points as an affine combination of the points in the set $S,$ if possible.Hint: Use Theorem 5 in Section 6.2 instead of row reduction to find the weights.
$$
\text { a. }\mathbf{p}_{1}=\left[\begin{array}{r}{0} \\ {-19} \\ {-5}\end{array}\right] \quad \text { b. } \mathbf{p}_{2}=\left[\begin{array}{r}{1.5} \\ {-1.3} \\ {-.5}\end{array}\right] \text { c. } \mathbf{p}_{3}=\left[\begin{array}{r}{5} \\ {-4} \\ {0}\end{array}\right]
$$

Gideon Idumah
Gideon Idumah
Numerade Educator
05:20

Problem 7

Let
$$\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {0} \\ {3} \\ {0}\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{r}{2} \\ {-1} \\ {0} \\ {4}\end{array}\right], \quad \mathbf{v}_{3}=\left[\begin{array}{r}{-1} \\ {2} \\ {1} \\ {1}\end{array}\right]$$
$$\mathbf{p}_{1}=\left[\begin{array}{r}{5} \\ {-3} \\ {5} \\ {3}\end{array}\right], \quad \mathbf{p}_{2}=\left[\begin{array}{r}{-9} \\ {10} \\ {9} \\ {-13}\end{array}\right], \quad \mathbf{p}_{3}=\left[\begin{array}{c}{4} \\ {2} \\ {8} \\ {5}\end{array}\right]$$
and $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} .$ It can be shown that $S$ is linearly independent.
a. Is $\mathbf{p}_{1}$ in Span $S ?$ Is $\mathbf{p}_{1}$ in aff $S ?$
b. Is $\mathbf{p}_{2}$ in Span $S ?$ Is $\mathbf{p}_{2}$ in aff $S ?$
c. Is $\mathbf{p}_{3}$ in Span $S ?$ Is $\mathbf{p}_{3}$ in aff $S ?$

Michael Jacobsen
Michael Jacobsen
Numerade Educator
05:51

Problem 8

Repeat Exercise 7 when
$$\mathbf{v}_{1}=\left[\begin{array}{r}{1} \\ {0} \\ {3} \\ {-2}\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{r}{2} \\ {1} \\ {6} \\ {-5}\end{array}\right], \quad \mathbf{v}_{3}=\left[\begin{array}{r}{3} \\ {0} \\ {12} \\ {-6}\end{array}\right]$$
$$\mathbf{p}_{1}=\left[\begin{array}{r}{4} \\ {-1} \\ {15} \\ {-7}\end{array}\right], \quad \mathbf{p}_{2}=\left[\begin{array}{r}{-5} \\ {3} \\ {-8} \\ {-6}\end{array}\right], \quad \text { and } \quad \mathbf{p}_{3}=\left[\begin{array}{r}{1} \\ {6} \\ {-6} \\ {-8}\end{array}\right]$$

Gideon Idumah
Gideon Idumah
Numerade Educator
02:12

Problem 9

Suppose that the solutions of an equation $A \mathbf{x}=\mathbf{b}$ are all of the form
$\mathbf{x}=x_{3} \mathbf{u}+\mathbf{p},$ where $\mathbf{u}=\left[\begin{array}{r}{4} \ {-2}\end{array}\right]$ and $\mathbf{p}=\left[\begin{array}{r}{-3} \\ {0}\end{array}\right]$ Find points $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ such that the solution set of $A \mathbf{x}=\mathbf{b}$ is
$\operatorname{aff}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$

Michael Jacobsen
Michael Jacobsen
Numerade Educator
03:01

Problem 10

Suppose that the solutions of an equation $A \mathbf{x}=\mathbf{b}$ are all of the form $\mathbf{x}=x_{3} \mathbf{u}+\mathbf{p},$ where $\mathbf{u}=\left[\begin{array}{r}{5} \\ {1} \\ {-2}\end{array}\right]$ and $\mathbf{p}=\left[\begin{array}{r}{1} \\ {-3} \\ {4}\end{array}\right]$ Find points $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ such that the solution set of $A \mathbf{x}=\mathbf{b}$ is aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$

Gideon Idumah
Gideon Idumah
Numerade Educator
01:51

Problem 11

In Exercises 11 and $12,$ mark each statement True or False. Justify each answer.
a. The set of all affine combinations of points in a set $S$ is called the affine hull of $S$ .
b. If $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}\right\}$ is a linearly independent subset of $\mathbb{R}^{n}$ and if $\mathbf{p}$ is a linear combination of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{k},$ then $\mathbf{p}$ is an affine combination of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}$
c. The affine hull of two distinct points is called a line.
d. A flat is a subspace.
e. A plane in $\mathbb{R}^{3}$ is a hyperplane.

LucĂ­a Guerrero
LucĂ­a Guerrero
Numerade Educator
02:50

Problem 12

In Exercises 11 and $12,$ mark each statement True or False. Justify each answer.
a. If $S=\{x\},$ then aff $S$ is the empty set.
b. A set is affine if and only if it contains its affine hull.
c. A flat of dimension 1 is called a line.
d. $A$ flat of dimension 2 is called a hyperplane.
e. A flat through the origin is a subspace.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:34

Problem 13

Suppose $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3} .$ Show that $\operatorname{Span}\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}\right\}$ is a plane in $\mathbb{R}^{3} .[\text { Hint: What can }$
you say about $\mathbf{u}$ and $\mathbf{v}$ when $\operatorname{Span}\{\mathbf{u}, \mathbf{v}\}$ is a plane? $]$

LucĂ­a Guerrero
LucĂ­a Guerrero
Numerade Educator
03:53

Problem 14

Show that if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3},$ then aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is the plane through $\mathbf{v}_{1}, \mathbf{v}_{2},$ and $\mathbf{v}_{3} .$

Gideon Idumah
Gideon Idumah
Numerade Educator
04:33

Problem 15

Let $A$ be an $m \times n$ matrix and, given $\mathbf{b}$ in $\mathbb{R}^{m},$ show that the
set $S$ of all solutions of $A \mathbf{x}=\mathbf{b}$ is an affine subset of $\mathbb{R}^{n}$ .

LucĂ­a Guerrero
LucĂ­a Guerrero
Numerade Educator
03:13

Problem 16

Let $\mathbf{v} \in \mathbb{R}^{n}$ and let $k \in \mathbb{R} .$ Prove that $S=\left\{\mathbf{x} \in \mathbb{R}^{n} : \mathbf{x} \cdot \mathbf{v}=k\right\}$ is an affine subset of $\mathbb{R}^{n}$ .

Gideon Idumah
Gideon Idumah
Numerade Educator
02:19

Problem 17

Choose a set $S$ of three points such that aff $S$ is the plane in $\mathbb{R}^{3}$ whose equation is $x_{3}=5 .$ Justify your work.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
03:50

Problem 18

Choose a set $S$ of four distinct points in $\mathbb{R}^{3}$ such that aff $S$ is the plane $2 x_{1}+x_{2}-3 x_{3}=12 .$ Justify your work.

Gideon Idumah
Gideon Idumah
Numerade Educator
04:12

Problem 19

Let $S$ be an affine subset of $\mathbb{R}^{n},$ suppose $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a linear transformation, and let $f(S)$ denote the set of images
$\{f(\mathbf{x}) : \mathbf{x} \in S\} .$ Prove that $f(S)$ is an affine subset of $\mathbb{R}^{m}$ .

Michael Jacobsen
Michael Jacobsen
Numerade Educator
04:11

Problem 20

Let $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation, let $T$ be an
affine subset of $\mathbb{R}^{m},$ and let $S=\left\{\mathbf{x} \in \mathbb{R}^{n} : f(\mathbf{x}) \in T\right\} .$ Show that $S$ is an affine subset of $\mathbb{R}^{n}$ .

Gideon Idumah
Gideon Idumah
Numerade Educator
02:54

Problem 21

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
If $A \subset B$ and $B$ is affine, then aff $A \subset B$ .

Michael Jacobsen
Michael Jacobsen
Numerade Educator
01:16

Problem 22

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
If $A \subset B,$ then aff $A \subset$ aff $B$

Gideon Idumah
Gideon Idumah
Numerade Educator
01:40

Problem 23

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
$[(\operatorname{aff} A) \cup(\operatorname{aff} B)] \subset \operatorname{aff}(A \cup B) .$ IHint: To show that $D \cup E \subset F,$ show that $D \subset F$ and $E \subset F . ]$

LucĂ­a Guerrero
LucĂ­a Guerrero
Numerade Educator
03:08

Problem 24

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
Find an example in $\mathbb{R}^{2}$ to show that equality need not hold in the statement of Exercise $23 .$ IHint: Consider sets $A$ and $B$ , each of which contains only one or two points. $]$

Gideon Idumah
Gideon Idumah
Numerade Educator
01:44

Problem 25

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
$\operatorname{aff}(A \cap B) \subset(\operatorname{aff} A \cap \operatorname{aff} B)$

LucĂ­a Guerrero
LucĂ­a Guerrero
Numerade Educator
04:52

Problem 26

In Exercises $21-26,$ prove the given statement about subsets $A$ and $B$ of $\mathbb{R}^{n},$ or provide the required example in $\mathbb{R}^{2} .$ A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).
Find an example in $\mathbb{R}^{2}$ to show that equality need not hold in the statement of Exercise $25 .$

Gideon Idumah
Gideon Idumah
Numerade Educator