A is Contained in the Set of All Linear Transformations Show That a is a Continuous Function

Problem 1

Sketch the image of the standard $L$ under the linear transformation
\[T(\vec{x})=\left[\begin{array}{ll}3 & 1 \\1 & 2\end{array}\right] \vec{x}\]
See Example 1.

Problem 2

Find the matrix of a rotatio$\left[\begin{array}{l}0 \\ 2\end{array}\right] .$ Let us calculate $T\left[\begin{array}{l}1 \\ 0\end{array}\right]$ and $T\left[\begin{array}{l}0 \\ 2\end{array}\right]$n through an angle of $60^{\circ}$ in the counterclockwise direction.

Problem 3

Consider a linear transformation $T$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$. Use $T\left(\vec{e}_{1}\right)$ and $T\left(\vec{e}_{2}\right)$ to describe the image of the unit square geometrically.

Problem 4

Interpret the following linear transformation geometrically:
\[T(\vec{x})=\left[\begin{array}{rr}1 & 1 \\-1 & 1\end{array}\right] \vec{x}\]

Problem 5

The matrix
\[\left[\begin{array}{rr}-0.8 & -0.6 \\0.6 & -0.8\end{array}\right]\]
represents a rotation. Find the angle of rotation (in radians)

Problem 6

Let $L$ be the line in $\mathbb{R}^{3}$ that consists of all scalar multiples of the vector $\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right] .$ Find the orthogonal projection of the vector $\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ onto $L$

Problem 7

Let $L$ be the line in $\mathbb{R}^{3}$ that consists of all scalar multiples of $\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right] .$ Find the reflection of the vector $\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ about the line $L$.

Problem 8

Interpret the following linear transformation geometrically:
\[T(\vec{x})=\left[\begin{array}{rr}0 & -1 \\-1 & 0\end{array}\right] \vec{x}\]

Problem 9

Interpret the following linear transformation geometrically:
\[T(\vec{x})=\left[\begin{array}{ll}1 & 0 \\1 & 1\end{array}\right] \vec{x}\]

Problem 10

Find the matrix of the orthogonal projection onto the line $L$ in $\mathbb{R}^{2}$ shown in the accompanying figure:

Problem 11

Refer to Exercise 10. Find the matrix of the reflection about the line $L$.

Problem 12

Consider a reflection matrix $A$ and a vector $\vec{x}$ in $\mathrm{R}^{2}$. We define $\vec{v}=\vec{x}+A \vec{x}$ and $\vec{w}=\vec{x}-A \vec{x}$
a. Using the definition of a reflection, express $A(A \vec{x})$ in terms of $\vec{x}$
b. Express $A \vec{v}$ in terms of $\vec{v}$
c. Express $A \vec{w}$ in terms of $\vec{w}$
d. If the vectors $\vec{v}$ and $\vec{w}$ are both nonzero, what is the angle between $\vec{v}$ and $\vec{w} ?$
e. If the vector $\vec{v}$ is nonzero, what is the relationship between $\vec{v}$ and the line $L$ of reflection?
Illustrate all parts of this exercise with a sketch showing $\vec{x}, A \vec{x}, A(A \vec{x}), \vec{v}, \vec{w},$ and the line $L$.

Problem 13

Suppose a line $L$ in $\mathbb{R}^{2}$ contains the unit vector
\[\vec{u}=\left[\begin{array}{l}u_{1} \\u_{2}\end{array}\right]\]
Find the matrix $A$ of the linear transformation $T(\vec{x})=\operatorname{ref}_{L}(\vec{x}) \cdot$ Give the entries of $A$ in terms of $u_{1}$ and $u_{2} .$ Show that $A$ is of the form $\left[\begin{array}{rr}a & b \\ b & -a\end{array}\right],$ where $a^{2}+b^{2}=1$.

Problem 14

Suppose a line $L$ in $\mathbb{R}^{3}$ contains the unit vector
\[\vec{u}=\left[\begin{array}{l}u_{1} \\u_{2} \\u_{3}\end{array}\right]\]
a. Find the matrix $A$ of the linear transformation $T(\vec{x})=\operatorname{proj}_{L}(\vec{x}) .$ Give the entries of $A$ in terms of the components $u_{1}, u_{2}, u_{3}$ of $\vec{u}$
b. What is the sum of the diagonal entries of the matrix $A$ you found in part (a)?

Problem 15

Suppose a line $L$ in $\mathbb{R}^{3}$ contains the unit vector
\[\vec{u}=\left[\begin{array}{l}u_{1} \\u_{2} \\u_{3}\end{array}\right]\]
Find the matrix $A$ of the linear transformation $T(\vec{x})=$ $\operatorname{ref}_{L}(\vec{x}) .$ Give the entries of $A$ in terms of the components $u_{1}, u_{2}, u_{3}$ of $\vec{u}$.

Problem 16

Let $T(\vec{x})=\operatorname{ref}_{L}(\vec{x})$ be the reflection about the line $L$ in
$\mathbb{R}^{2}$ shown in the accompanying figure.
a. Draw sketches to illustrate that $T$ is linear.
b. Find the matrix of $T$ in terms of $\theta$

Ekaveera Kumar

Numerade Educator

Problem 17

Consider a matrix $A$ of the form $A=\left[\begin{array}{rr}a & b \\ b & -a\end{array}\right],$ where $a^{2}+b^{2}=1 .$ Find two nonzero perpendicular vectors $\vec{v}$ and $\vec{w}$ such that $A \vec{v}=\vec{v}$ and $A \vec{w}=-\vec{w}$ (write the entries of $\vec{v}$ and $\vec{w}$ in terms of $a$ and $b$ ). Conclude that $T(\vec{x})=A \vec{x}$ represents the reflection about the line $L$ spanned by $\vec{v}$.

Problem 18

The linear transformation $T(\vec{x})=\left[\begin{array}{rr}0.6 & 0.8 \\ 0.8 & -0.6\end{array}\right] \vec{x}$ is a reflection about a line $L$. See Exercise 17 . Find the equation of line $L$ (in the form $y=m x$ ).

Problem 19

Find the matrices of the linear transformations from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ given in Exercises 19 through $23 .$ Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear.
The orthogonal projection onto the $x-y$ -plane.

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Rotations and reflections have two remarkable properties: They preserve the length of vectors and the angle between vectors. (Draw figures illustrating these properties.) We will show that, conversely, any linear transformation $T$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ that preserves length and angles is either a rotation or a reflection (about a line).
a. Show that if $T(\vec{x})=A \vec{x}$ preserves length and angles, then the two column vectors $\vec{v}$ and $\vec{w}$ of $A$ must be perpendicular unit vectors.
b. Write the first column vector of $A$ as $\vec{v}=\left[\begin{array}{l}a \\ b\end{array}\right] ;$ note that $a^{2}+b^{2}=1,$ since $\vec{v}$ is a unit vector. Show that for a given $\vec{v}$ there are two possibilities for $\vec{w},$ the second column vector of $A .$ Draw a sketch showing $\vec{v}$ and the two possible vectors $\vec{w} .$ Write the components of $\vec{w}$ in terms of $a$ and $b$
c. Show that if a linear transformation $T$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ preserves length and angles, then $T$ is either a rotation or a reflection (about a line). See Exercise 17.

Problem 25

Find the inverse of the matrix $\left[\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right],$ where $k$ is an arbitrary constant. Interpret your result geometrically.

Problem 26

a. Find the scaling matrix $A$ that transforms $\left[\begin{array}{r}2 \\ -1\end{array}\right]$ into $\left[\begin{array}{r}8 \\ -4\end{array}\right]$
b. Find the orthogonal projection matrix $B$ that transforms $\left[\begin{array}{l}2 \\ 3\end{array}\right]$ into $\left[\begin{array}{l}2 \\ 0\end{array}\right]$
c. Find the rotation matrix $C$ that transforms $\left[\begin{array}{l}0 \\ 5\end{array}\right]$ into $\left[\begin{array}{l}3 \\ 4\end{array}\right]$
d. Find the shear matrix $D$ that transforms $\left[\begin{array}{l}1 \\ 3\end{array}\right]$ into $\left[\begin{array}{l}7 \\ 3\end{array}\right]$
e. Find the reflection matrix $E$ that transforms $\left[\begin{array}{l}7 \\ 1\end{array}\right]$ into $\left[\begin{array}{r}-5 \\ 5\end{array}\right]$

Problem 27

Consider the matrices $A$ through $E$ below.
\[\begin{aligned}A=\left[\begin{array}{rr}0.6 & 0.8 \\0.8 & -0.6\end{array}\right], \quad B=\left[\begin{array}{ll}3 & 0 \\0 & 3\end{array}\right] \\C=\left[\begin{array}{rr}0.36 & -0.48 \\
-0.48 & 0.64\end{array}\right], \quad D=\left[\begin{array}{rr}-0.8 & 0.6 \\-0.6 & -0.8\end{array}\right] \\
E=\left[\begin{array}{rr}1 & 0 \\-1 & 1\end{array}\right]\end{aligned}\]
Fill in the blanks in the sentences below. We are told that there is a solution in each case.
Matrix ___ represents a scaling.
Matrix ___ represents an orthogonal projection.
Matrix ___ represents a shear.
Matrix ___ represents a reflection.
Matrix ___ represents a rotation.

Problem 28

Each of the linear transformations in parts (a) through
(e) corresponds to one (and only one) of the matrices $A$ through $J$. Match them up.
a. Scaling
b. Shear
$\mathbf{c}_{*} \quad$ Rotation
d. Orthogonal projection e. $\quad$ Reflection $A=\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right], \quad B=\left[\begin{array}{ll}2 & 1 \\ 1 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}-0.6 & 0.8 \\ -0.8 & -0.6\end{array}\right]$
$D=\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right], \quad E=\left[\begin{array}{rr}1 & 0 \\ -3 & 1\end{array}\right], \quad F=\left[\begin{array}{rr}0.6 & 0.8 \\ 0.8 & -0.6\end{array}\right]$
$\begin{aligned} G=\left[\begin{array}{ll}0.6 & 0.6 \\ 0.8 & 0.8\end{array}\right], \quad H=\left[\begin{array}{rr}2 & -1 \\ 1 & 2\end{array}\right], \quad I=\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right] \\ J &=\left[\begin{array}{ll}0.8 & -0.6 \\ 0.6 & -0.8\end{array}\right] \end{aligned}$

Problem 29

Let $T$ be a function from $\mathbb{R}^{m}$ to $\mathbb{R}^{n},$ and let $L$ be a function from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$. Suppose that $L(T(\vec{x}))=\vec{x}$ for all $\vec{x}$ in $\mathbb{R}^{m}$ and $T(L(\vec{y}))=\vec{y}$ for all $\vec{y}$ in $\mathbb{R}^{n} .$ If $T$ is a linear transformation, show that $L$ is linear as well. Hint: $\vec{v}+\vec{w}=T(L(\vec{v}))+T(L(\vec{w}))=T(L(\vec{v})+L(\vec{w}))$
since $T$ is linear. Now apply $L$ on both sides.

Problem 30

Find a nonzero $2 \times 2$ matrix $A$ such that $A \vec{x}$ is parallel to the vector $\left[\begin{array}{l}1 \\ 2\end{array}\right],$ for all $\vec{x}$ in $\mathbb{R}^{2}$.

Problem 31

Find a nonzero $3 \times 3$ matrix $A$ such that $A \vec{x}$ is perpendicular to $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right],$ for all $\vec{x}$ in $\mathbb{R}^{3}$.

Problem 32

Consider the rotation matrix $D=\left[\begin{array}{cr}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
and the vector $\vec{v}=\left[\begin{array}{l}\cos \beta \\ \sin \beta\end{array}\right],$ where $\alpha$ and $\beta$ are arbitrary angles.
a. Draw a sketch to explain why $D \vec{v}=\left[\begin{array}{c}\cos (\alpha+\beta) \\ \sin (\alpha+\beta)\end{array}\right]$
b. Compute $D \vec{v}$. Use the result to derive the addition theorems for sine and cosine:
\[\cos (\alpha+\beta)=\ldots ., \quad \sin (\alpha+\beta)=\ldots\]

Problem 33

Consider two nonparallel lines $L_{1}$ and $L_{2}$ in $\mathbb{R}^{2}$. Explain why a vector $\vec{v}$ in $\mathbb{R}^{2}$ can be expressed uniquely as
\[\vec{v}=\vec{v}_{1}+\vec{v}_{2}\]
where $\vec{v}_{1}$ is on $L_{1}$ and $\vec{v}_{2}$ on $L_{2}$. Draw a sketch. The transformation $T(\vec{v})=\vec{v}_{1}$ is called the projection onto $L_{1}$ along $L_{2}$. Show algebraically that $T$ is linear.

Problem 34

One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about a line. Identify both and briefly justify your choice.
$$\begin{aligned}A=\frac{1}{3}\left[\begin{array}{rrr}1 & 2 & 2 \\2 & 1 & 2 \\
2 & 2 & 1\end{array}\right], \quad B=\frac{1}{3}\left[\begin{array}{rrr}1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1\end{array}\right] \\
C=\frac{1}{3}\left[\begin{array}{rrr}2 & 1 & 1 \\1 & 2 & 1 \\1 & 1 & 2\end{array}\right], \quad D=-\frac{1}{3}\left[\begin{array}{rrr}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1\end{array}\right] \\E &=\frac{1}{3}\left[\begin{array}{rrr}-1 & 2 & 2 \\2 & -1 & 2 \\2 & 2 & -1
\end{array}\right]\end{aligned}$$

Problem 35

Let $T$ be an invertible linear transformation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2} .$ Let $P$ be a parallelogram in $\mathbb{R}^{2}$ with one vertex at the origin. Is the image of $P$ a parallelogram as well? Explain. Draw a sketch of the image.

Problem 36

Let $T$ be an invertible linear transformation from $\mathbb{R}^{2}$ to $\mathrm{R}^{2}$. Let $P$ be a parallelogram in $\mathrm{R}^{2}$. Is the image of $P$ a parallelogram as well? Explain.

Problem 37

The trace of a matrix $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is the sum $a+d$ of its diagonal entries. What can you say about the trace of a $2 \times 2$ matrix that represents a(n)
a. orthogonal projection
b. reflection about a line
c. rotation
d. (horizontal or vertical) shear.
In three cases, give the exact value of the trace, and in one case, give an interval of possible values.

Problem 38

The determinant of a matrix $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is $a d-b c$ (we have seen this quantity in Exercise 2.1 .13 already). Find the determinant of a matrix that represents a(n)
a. orthogonal projection
b. reflection about a line
$\mathbf{c}$. rotation
d. (horizontal or vertical) shear. What do your answers tell you about the invertibility of these matrices?

Problem 39

Describe each of the linear transformations defined by the matrices in parts (a) through (c) geometrically, as a well-known transformation combined with a scaling. Give the scaling factor in each case.
a. $\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
b. $\left[\begin{array}{rr}3 & 0 \\ -1 & 3\end{array}\right]$
$c_{*}\left[\begin{array}{rr}3 & 4 \\ 4 & -3\end{array}\right]$

Problem 40

Let $P$ and $Q$ be two perpendicular lines in $\mathbb{R}^{2}$. For a vector $\vec{x}$ in $\mathbb{R}^{2},$ what is proj $_{P}(\vec{x})+\operatorname{proj}_{Q}(\vec{x}) ?$ Give your answer in terms of $\vec{x}$. Draw a sketch to justify your answer.

Problem 41

Let $P$ and $Q$ be two perpendicular lines in $\mathbb{R}^{2}$. For a vector $\vec{x}$ in $\mathbb{R}^{2},$ what is the relationship between $\operatorname{ref}_{P}(\vec{x})$ and $\operatorname{ref}_{Q}(\vec{x}) ?$ Draw a sketch to justify your answer.

Problem 42

Let $T(\vec{x})=\operatorname{proj}_{L}(\vec{x})$ be the orthogonal projection onto a line in $\mathbb{R}^{2}$. What is the relationship between $T(\vec{x})$ and $T(T(\vec{x})) ?$ Justify your answer carefully.

Problem 43

Use the formula derived in Exercise 2.1 .13 to find the inverse of the rotation matrix
\[A=\left[\begin{array}{lr}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta\end{array}\right]\]
Interpret the linear transformation defined by $A^{-1}$ geometrically. Explain.

Problem 44

A nonzero matrix of the form $A=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]$ repre sents a rotation combined with a scaling. Use the formula derived in Exercise 2.1 .13 to find the inverse of $A$. Interpret the linear transformation defined by $A^{-1}$ geometrically. Explain.

Problem 45

A matrix of the form $A=\left[\begin{array}{cc}a & b \\ b & -a\end{array}\right],$ where $a^{2}+b^{2}=$
1, represents a reflection about a line. See Exercise 17 Use the formula derived in Exercise 2.1 .13 to find the inverse of $A$. Explain.

Problem 46

A nonzero matrix of the form $A=\left[\begin{array}{rr}a & b \\ b & -a\end{array}\right]$ repre sents a reflection about a line $L$ combined with a scaling. (Why? What is the scaling factor?) Use the formula derived in Exercise 2.1 .13 to find the inverse of $A$. Inter pret the linear transformation defined by $A^{-1}$ geometrically. Explain.

Problem 47

In this exercise we will prove the following remarkable theorem: If $T(\vec{x})=A \vec{x}$ is any linear transfor mation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2},$ then there exist perpendicular unit vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ in $\mathbb{R}^{2}$ such that the vectors $T\left(\vec{v}_{1}\right)$ and $T\left(\vec{v}_{2}\right)$ are perpendicular as well (see the accompanying figure $),$ in the sense that $T\left(\vec{v}_{1}\right) \cdot T\left(\vec{v}_{2}\right)=$
0. This is not intuitively obvious: Think about the case of a shear, for example. For a generalization, see Theorem 8.3 .3

For any real number $t,$ the vectors $\left[\begin{array}{c}\cos t \\ \sin t\end{array}\right]$ and $\left[\begin{array}{c}-\sin t \\ \cos t\end{array}\right]$ will be perpendicular unit vectors. Now we
can consider the function
\[\begin{aligned}f(t) &=\left(T\left[\begin{array}{c}\cos t \\\sin t\end{array}\right]\right) \cdot\left(T\left[\begin{array}{c}-\sin t \\
\cos t\end{array}\right]\right) \\&=\left(A\left[\begin{array}{c}\cos t \\
\sin t
\end{array}\right]\right) \cdot\left(A\left[\begin{array}{c}-\sin t \\\cos t\end{array}\right]\right)\end{aligned}\]
It is our goal to show that there exists a number $c$ such that $f(c)=\left(T\left[\begin{array}{c}\cos c \\ \sin c\end{array}\right]\right) \cdot\left(T\left[\begin{array}{c}-\sin c \\ \cos c\end{array}\right]\right)=0 .$ Then
the vectors $\vec{v}_{1}=\left[\begin{array}{c}\cos c \\ \sin c\end{array}\right]$ and $\vec{v}_{2}=\left[\begin{array}{c}-\sin c \\ \cos c\end{array}\right]$ will
have the required property that they are perpendicular unit vectors such that $T\left(\vec{v}_{1}\right) \cdot T\left(\vec{v}_{2}\right)=0$
a. Show that the function $f(t)$ is continuous. You may assume that $\cos t, \sin t,$ and constant functions are continuous. Also, sums and products of continuous functions are continuous. Hint: Write $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
b. Show that $f\left(\frac{\pi}{2}\right)=-f(0)$
c. Show that there exists a number $c,$ with $0 \leq c \leq \frac{\pi}{2}$ such that $f(c)=0 .$ Hint: Use the intermediate value theorem: If a function $f(t)$ is continuous for $a \leq t \leq b$ and if $L$ is any number between $f(a)$ and $f(b),$ then there exists a number $c$ between $a$ and $b$ with $f(c)=L$.

Problem 48

If $a 2 \times 2$ matrix $A$ represents a rotation, find perpendicular unit vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ in $\mathbb{R}^{2}$ such that the vectors $T\left(\vec{v}_{1}\right)$ and $T\left(\vec{v}_{2}\right)$ are perpendicular as well. See Exercise 47.

Problem 49

For the linear transformations $T$ in Exercises 49 through $52,$ do the following:
a. Find the function $f(t)$ defined in Exercise 47 and graph it for $0 \leq t \leq \frac{\pi}{2},$ You may use technology.
b. Find a number $c,$ with $0 \leq c \leq \frac{\pi}{2},$ such that $f(c)=0 .(\text {In Problem } 50,$ approximate c to three
significamt digits, using technology.
c. Find perpendicular unit vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ in $\mathbb{R}^{2}$ such that the vectors $T\left(\vec{v}_{1}\right)$ and $T\left(\vec{v}_{2}\right)$ are perpendicular as well. Draw a sketch showing $\vec{v}_{1}, \vec{v}_{2}$
$T\left(\vec{v}_{1}\right),$ and $T\left(\vec{v}_{2}\right)$.
$$T(\vec{x})=\left[\begin{array}{cc}2 & 2 \\1 & -4\end{array}\right] \vec{x}$$

Problem 50

Problem 51

Problem 52

Problem 53

Sketch the image of the unit circle under the linear transformation
\[T(\vec{x})=\left[\begin{array}{ll}5 & 0 \\0 & 2\end{array}\right] \vec{x}\]

Problem 54

Let $T$ be an invertible linear transformation from $\mathbb{R}^{2}$ to $\mathrm{R}^{2} .$ Show that the image of the unit circle is an ellipse centered at the origin. $^{8}$ Hint: Consider two perpendicular unit vectors $\vec{v}_{1}$ and $\vec{v}_{2}$ such that $T\left(\vec{v}_{1}\right)$ and $T\left(\vec{v}_{2}\right)$ are perpendicular. See Exercise $47 .$ The unit circle consists of all vectors of the form
\[\vec{v}=\cos (t) \vec{v}_{1}+\sin (t) \vec{v}_{2}\]
where $t$ is a parameter.

Problem 55

Let $\vec{w}_{1}$ and $\vec{w}_{2}$ be two nonparallel vectors in $\mathbb{R}^{2}$. Consider the curve $C$ in $\mathbb{R}^{2}$ that consists of all vectors of the form $\cos (t) \vec{w}_{1}+\sin (t) \vec{w}_{2},$ where $t$ is a parameter.
Show that $C$ is an ellipse. Hint: You can interpret $C$ as the image of the unit circle under a suitable linear transformation; then use Exercise 54.

Problem 56

Consider an invertible linear transformation $T$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$. Let $C$ be an ellipse in $\mathbb{R}^{2}$. Show that the image of $C$ under $T$ is an ellipse as well. Hint: Use the result of Exercise 55.

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Source: https://www.numerade.com/books/chapter/linear-transformations-9/?section=23683