Linear Algebra and Its Applications

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

Chapter 4

Vector Spaces - all with Video Answers

Educators

Section 4

Coordinate Systems

Problem 1

In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{3} \\ {-5}\end{array}\right],\left[\begin{array}{r}{-4} \\ {6}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{l}{5} \\ {3}\end{array}\right]
$$

Michael J.

Michael J.

Numerade Educator

Problem 2

In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{l}{4} \\ {5}\end{array}\right],\left[\begin{array}{l}{6} \\ {7}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{8} \\ {-5}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 3

In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{1} \\ {-4} \\ {3}\end{array}\right],\left[\begin{array}{r}{5} \\ {2} \\ {-2}\end{array}\right],\left[\begin{array}{r}{4} \\ {-7} \\ {0}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{3} \\ {0} \\ {-1}\end{array}\right]
$$

Michael J.

Numerade Educator

Problem 4

In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{-1} \\ {2} \\ {0}\end{array}\right],\left[\begin{array}{r}{3} \\ {-5} \\ {2}\end{array}\right],\left[\begin{array}{r}{4} \\ {-7} \\ {3}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{-4} \\ {8} \\ {-7}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 5

In Exercises $5-8,$ find the coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ of $\mathbf{x}$ relative to the given basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$
$$
\mathbf{b}_{1}=\left[\begin{array}{r}{1} \\ {-3}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{2} \\ {-5}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{-2} \\ {1}\end{array}\right]
$$

Michael J.

Numerade Educator

Problem 6

In Exercises $5-8,$ find the coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ of $\mathbf{x}$ relative to the given basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$
$$
\mathbf{b}_{1}=\left[\begin{array}{r}{1} \\ {-2}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{5} \\ {-6}\end{array}\right], \mathbf{x}=\left[\begin{array}{l}{4} \\ {0}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 7

In Exercises $5-8,$ find the coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ of $\mathbf{x}$ relative to the given basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$
$$
\mathbf{b}_{1}=\left[\begin{array}{r}{1} \\ {-1} \\ {-3}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{-3} \\ {4} \\ {9}\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}{2} \\ {-2} \\ {4}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{8} \\ {-9} \\ {6}\end{array}\right]
$$

Priyanka S.

Numerade Educator

Problem 8

In Exercises $5-8,$ find the coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ of $\mathbf{x}$ relative to the given basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$
$$
\mathbf{b}_{1}=\left[\begin{array}{l}{1} \\ {0} \\ {3}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{l}{2} \\ {1} \\ {8}\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}{1} \\ {-1} \\ {2}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{3} \\ {-5} \\ {4}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 9

In Exercises 9 and $10,$ find the change-of-coordinates matrix from $\mathcal{B}$ to the standard basis in $\mathbb{R}^{n} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{2} \\ {-9}\end{array}\right],\left[\begin{array}{l}{1} \\ {8}\end{array}\right]\right\}
$$

Priyanka S.

Numerade Educator

Problem 10

In Exercises 9 and $10,$ find the change-of-coordinates matrix from $\mathcal{B}$ to the standard basis in $\mathbb{R}^{n} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{3} \\ {-1} \\ {4}\end{array}\right],\left[\begin{array}{r}{2} \\ {0} \\ {-5}\end{array}\right],\left[\begin{array}{r}{8} \\ {-2} \\ {7}\end{array}\right]\right\}
$$

Doruk I.

Numerade Educator

Problem 11

In Exercises 11 and $12,$ use an inverse matrix to find $[\mathbf{x}]_{\mathcal{B}}$ for the given $\mathbf{x}$ and $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{3} \\ {-5}\end{array}\right],\left[\begin{array}{r}{-4} \\ {6}\end{array}\right]\right\}, \mathbf{x}=\left[\begin{array}{r}{2} \\ {-6}\end{array}\right]
$$

Priyanka S.

Numerade Educator

Problem 12

In Exercises 11 and $12,$ use an inverse matrix to find $[\mathbf{x}]_{\mathcal{B}}$ for the given $\mathbf{x}$ and $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{l}{4} \\ {5}\end{array}\right],\left[\begin{array}{l}{6} \\ {7}\end{array}\right]\right\}, \mathbf{x}=\left[\begin{array}{l}{2} \\ {0}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 13

The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ Find the coordinate vector of $\mathbf{p}(t)=1+4 t+7 t^{2}$ relative to $\mathcal{B} .$

Priyanka S.

Numerade Educator

Problem 14

The set $\mathcal{B}=\left\{1-t^{2}, t-t^{2}, 2-2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ . Find the coordinate vector of $\mathbf{p}(t)=3+t-6 t^{2}$ relative to $\mathcal{B} .$

Doruk I.

Numerade Educator

Problem 15

In Exercises 15 and $16,$ mark each statement True or False. Justify each answer. Unless stated otherwise, $\mathcal{B}$ is a basis for a vector space $V .$
a. If $\mathbf{x}$ is in $V$ and if $\mathcal{B}$ contains $n$ vectors, then the $\mathcal{B}$ -
coordinate vector of $\mathbf{x}$ is in $\mathbb{R}^{n}$ .
b. If $P_{\mathcal{B}}$ is the change-of-coordinates matrix, then $[\mathbf{x}]_{\mathcal{B}}=$ $P_{\mathcal{B}} \mathbf{x},$ for $\mathbf{x}$ in $V$ .
c. The vector spaces $\mathbb{P}_{3}$ and $\mathbb{R}^{3}$ are isomorphic.

Priyanka S.

Numerade Educator

Problem 16

In Exercises 15 and $16,$ mark each statement True or False. Justify each answer. Unless stated otherwise, $\mathcal{B}$ is a basis for a vector space $V .$
a. If $\mathcal{B}$ is the standard basis for $\mathbb{R}^{n},$ then the $\mathcal{B}$ -coordinate vector of an $\mathbf{x}$ in $\mathbb{R}^{n}$ is $\mathbf{x}$ itself.
b. The correspondence $[\mathbf{x}]_{\mathcal{B}} \mapsto \mathbf{x}$ is called the coordinate mapping.
c. In some cases, a plane in $\mathbb{R}^{3}$ can be isomorphic to $\mathbb{R}^{2}$ .

Doruk I.

Numerade Educator

Problem 17

The vectors $\mathbf{v}_{1}=\left[\begin{array}{r}{1} \\ {-3}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{2} \\ {-8}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{-3} \\ {7}\end{array}\right] \operatorname{span} \mathbb{R}^{2}$ but do not form a basis. Find two different ways to express $\left[\begin{array}{l}{1} \\ {1}\end{array}\right]$ as a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$

Priyanka S.

Numerade Educator

Problem 18

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ be a basis for a vector space $V .$ Explain why the $\mathcal{B}$ -coordinate vectors of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}$ are the columns $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ of the $n \times n$ identity matrix.

Doruk I.

Numerade Educator

Problem 19

Let $S$ be a finite set in a vector space $V$ with the property that every $x$ in $V$ has a unique representation as a linear combination of elements of $S .$ Show that $S$ is a basis of $V .$

Priyanka S.

Numerade Educator

Problem 20

Suppose $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}\right\}$ is a linearly dependent spanning set for a vector space $V .$ Show that each $\mathbf{w}$ in $V$ can be expressed in more than one way as a linear combination of $\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}$ . [Hint: Let $\mathbf{w}=k_{1} \mathbf{v}_{1}+\cdots+k_{4} \mathbf{v}_{4}$ be an arbitrary vector in $V$ Use the linear dependence of $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}\right\}$ to produce another representation of $\mathbf{w}$ as a linear combination of $\mathbf{v}_{1}, \ldots, \mathbf{v}_{4} . ]$

Doruk I.

Numerade Educator

Problem 21

Let $\mathcal{B}=\left\{\left[\begin{array}{r}{1} \\ {-4}\end{array}\right],\left[\begin{array}{r}{-2} \\ {9}\end{array}\right]\right\} .$ since the coordinate mapping determined by $\mathcal{B}$ is a linear transformation from $\mathbb{R}^{2}$ into $\mathbb{R}^{2}$ , this mapping must be implemented by some $2 \times 2$ matrix $A$ . Find it. IHint: Multiplication by $A$ should transform a vector $\mathbf{x}$ into its coordinate vector $[\mathbf{x}]_{\mathcal{B}} . ]$

Priyanka S.

Numerade Educator

Problem 22

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ be a basis for $\mathbb{R}^{n}$ . Produce a description of an $n \times n$ matrix $A$ that implements the coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}} .$ (See Exercise $21 . )$

Doruk I.

Numerade Educator

Problem 23

Exercises $23-26$ concern a vector space $V,$ a basis $\mathcal{B}=$ $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\},$ and the coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}$
Show that the coordinate mapping is one-to-one. [Hint: Suppose $[\mathbf{u}]_{\mathcal{B}}=[\mathbf{w}]_{\mathcal{B}}$ for some $\mathbf{u}$ and $\mathbf{w}$ in $V,$ and show that $\mathbf{u}=\mathbf{w} . ]$

Priyanka S.

Numerade Educator

Problem 24

Exercises $23-26$ concern a vector space $V,$ a basis $\mathcal{B}=$ $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\},$ and the coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}$
Show that the coordinate mapping is onto $\mathbb{R}^{n} .$ That is, given any $\mathbf{y}$ in $\mathbb{R}^{n},$ with entries $y_{1}, \ldots, y_{n},$ produce $\mathbf{u}$ in $V$ such that $[\mathbf{u}]_{\mathcal{B}}=\mathbf{y} .$

Doruk I.

Numerade Educator

Problem 25

Exercises $23-26$ concern a vector space $V,$ a basis $\mathcal{B}=$ $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\},$ and the coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}$
Show that a subset $\left\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}\right\} \quad$ in $V$ is linearly
independent if and only if the set of coordinate vectors $\left\{\left[\mathbf{u}_{1}\right]_{\mathcal{B}}, \ldots,\left[\mathbf{u}_{p}\right]_{\mathcal{B}}\right\}$ is linearly independent in $\mathbb{R}^{n} .[\text { Hint: }$ since the coordinate mapping is one-to-one, the following equations have the same solutions, $c_{1}, \ldots, c_{p} . ]$

Priyanka S.

Numerade Educator

Problem 26

Exercises $23-26$ concern a vector space $V,$ a basis $\mathcal{B}=$ $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\},$ and the coordinate mapping $\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}$
Given vectors $\mathbf{u}_{1}, \ldots, \mathbf{u}_{p},$ and $\mathbf{w}$ in $V,$ show that $\mathbf{w}$ is a linear combination of $\mathbf{u}_{1}, \ldots, \mathbf{u}_{p}$ if and only if $[\mathbf{w}]_{\mathcal{B}}$ is a linear combination of the coordinate vectors $\left[\mathbf{u}_{1}\right]_{\mathcal{B}}, \ldots,\left[\mathbf{u}_{p}\right]_{\mathcal{B}}$

Doruk I.

Numerade Educator

Problem 27

In Exercises $27-30$ , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
$$
1+2 t^{3}, 2+t-3 t^{2},-t+2 t^{2}-t^{3}
$$

Priyanka S.

Numerade Educator

Problem 28

In Exercises $27-30$ , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
$$
1-2 t^{2}-t^{3}, t+2 t^{3}, 1+t-2 t^{2}
$$

Doruk I.

Numerade Educator

Problem 29

In Exercises $27-30$ , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
$$
(1-t)^{2}, t-2 t^{2}+t^{3},(1-t)^{3}
$$

Priyanka S.

Numerade Educator

Problem 30

In Exercises $27-30$ , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
$$
(2-t)^{3},(3-t)^{2}, 1+6 t-5 t^{2}+t^{3}
$$

Doruk I.

Numerade Educator

Problem 31

Use coordinate vectors to test whether the following sets of polynomials span $\mathbb{P}_{2}$ . Justify your conclusions.
a. $1-3 t+5 t^{2},-3+5 t-7 t^{2},-4+5 t-6 t^{2}, 1-t^{2}$
b. $5 t+t^{2}, 1-8 t-2 t^{2},-3+4 t+2 t^{2}, 2-3 t$

Priyanka S.

Numerade Educator

Problem 32

Let $\mathbf{p}_{1}(t)=1+t^{2}, \mathbf{p}_{2}(t)=t-3 t^{2}, \mathbf{p}_{3}(t)=1+t-3 t^{2}$
a. Use coordinate vectors to show that these polynomials form a basis for $\mathbb{P}_{2} .$
b. Consider the basis $\mathcal{B}=\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$ for $\mathbb{P}_{2} .$ Find $\mathbf{q}$ in $\mathbb{P}_{2}$ given that $[\mathbf{q}]_{\mathcal{B}}=\left[\begin{array}{r}{-1} \\ {1} \\ {2}\end{array}\right]$

Doruk I.

Numerade Educator

Problem 33

In Exercises 33 and $34,$ determine whether the sets of polynomials form a basis for $\mathbb{P}_{3}$ . Justify your conclusions.
$$
[\mathbf{M}] 3+7 t, 5+t-2 t^{3}, t-2 t^{2}, 1+16 t-6 t^{2}+2 t^{3}
$$

Priyanka S.

Numerade Educator

Problem 34

In Exercises 33 and $34,$ determine whether the sets of polynomials form a basis for $\mathbb{P}_{3}$ . Justify your conclusions.
$$
[\mathbf{M}] 5-3 t+4 t^{2}+2 t^{3}, 9+t+8 t^{2}-6 t^{3}, 6-2 t+5 t^{2}, t^{3}
$$

Doruk I.

Numerade Educator

Problem 35

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ and $\mathcal{B}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\} .$ Show that $\mathbf{x}$ is in $H$ and find the $\mathcal{B}$ -coordinate vector of $\mathbf{x},$ for
$$
\mathbf{v}_{1}=\left[\begin{array}{r}{11} \\ {-5} \\ {10} \\ {7}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{c}{14} \\ {-8} \\ {13} \\ {10}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{19} \\ {-13} \\ {18} \\ {15}\end{array}\right]
$$

Priyanka S.

Numerade Educator

Problem 36

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ and $\mathcal{B}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} .$ Show that $\mathcal{B}$ is a basis for $H$ and $\mathbf{x}$ is in $H,$ and find the $\mathcal{B}$ -coordinate vector of $\mathbf{x},$ for
$$
\mathbf{v}_{1}=\left[\begin{array}{r}{-6} \\ {4} \\ {-9} \\ {4}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{8} \\ {-3} \\ {7} \\ {-3}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{-9} \\ {5} \\ {-8} \\ {3}\end{array}\right], \mathbf{x}=\left[\begin{array}{r}{4} \\ {7} \\ {-8} \\ {3}\end{array}\right]
$$

Doruk I.

Numerade Educator

Problem 37

[M] Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying figure. The vectors $\left[\begin{array}{r}{2.6} \\ {-1.5} \\ {0}\end{array}\right],\left[\begin{array}{c}{0} \\ {3} \\{0}\end{array}\right],\left[\begin{array}{c}{0} \\ {0} \\ {4.8}\end{array}\right]$ in $\mathbb{R}^{3}$ form a basis for the unit cell shown on the right. The numbers here are Ängstrom units $\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right) .$ In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedral sites (so named because of the geometric objects formed by atoms at these locations).
Graph cannot copy
The hexagonal close-packed lattice and its unit cell.
One of the octahedral sites is $\left[\begin{array}{c}{1 / 2} \\ {1 / 4} \\ {1 / 6}\end{array}\right],$ relative to the lattice basis. Determine the coordinates of this site relative to the standard basis of $\mathbb{R}^{3}$ .

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Problem 38

[M] Exercises 37 and 38 concern the crystal lattice for titanium, which has the hexagonal structure shown on the left in the accompanying figure. The vectors $\left[\begin{array}{r}{2.6} \\ {-1.5} \\ {0}\end{array}\right],\left[\begin{array}{c}{0} \\ {3} \\{0}\end{array}\right],\left[\begin{array}{c}{0} \\ {0} \\ {4.8}\end{array}\right]$ in $\mathbb{R}^{3}$ form a basis for the unit cell shown on the right. The numbers here are Ängstrom units $\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right) .$ In alloys of titanium, some additional atoms may be in the unit cell at the octahedral and tetrahedral sites (so named because of the geometric objects formed by atoms at these locations).
Graph cannot copy
The hexagonal close-packed lattice and its unit cell.
One of the tetrahedral sites is $\left[\begin{array}{c}{1 / 2} \\ {1 / 2} \\ {1 / 3}\end{array}\right] .$ Determine the coordinates of this site relative to the standard basis of $\mathbb{R}^{3}$ .

Doruk I.

Numerade Educator