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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q23E (page 119)

Describe the images and kernels of the transformations in Exercises23through 25 geometrically.
23. Reflection about the line $y = \frac{x}{3} in {鈩�}^{2} {鈩�}^{2}$ .

Short Answer

Expert verified

The kernel is $\{\overset{鈫�}{0}\}$ , image is all of ${鈩�}^{2} {鈩�}^{2}$ .

Step by step solution

01

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$i m a g e (f) = \{f (x) : x i n X\} = \{b i n Y : b = f (x), for some x in X\}$

The Kernel of the transformation results in $\{\overset{鈫�}{0}\}$ , as the reflection is its own inverse.

As the inverse of the reflection exists thus, the image will be ${鈩�}^{2}$ . The kernel is of dimension $\{\overset{鈫�}{0}\}$ so, the image will be of ${鈩�}^{2}$ .

02

Final answer.

The kernel is $\{\overset{鈫�}{0}\}$ , image is all of ${鈩�}^{2}$ .

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Most popular questions from this chapter

(a) Let $V$ be a subset of role="math" localid="1660109056998" ${鈩�}^{n}$ . Let $m$ be the largest number of linearly independent vectors we can find in $V$ . (Note $m 鈮� n$ , by Theorem 3.2.8.) Choose linearly independent vectors ${\overset{鈫�}{蠀}}_{_{1}}, 鈥� 鈥� {\overset{鈫�}{蠀}}_{_{2}}, 鈥� 鈥�, 鈥� 鈥� {\overset{鈫�}{蠀}}_{_{m}}$ in $V$ . Show that the vectors ${\overset{鈫�}{蠀}}_{_{1}}, 鈥� 鈥� {\overset{鈫�}{蠀}}_{_{2}}, 鈥� 鈥�, 鈥� 鈥� {\overset{鈫�}{蠀}}_{_{m}}$ span $V$ and are therefore a basis of $V$ . This exercise shows that any subspace of ${鈩�}^{n}$ has a basis.

If you are puzzled, think first about the special case when role="math" localid="1660109086728" $V$ is a plane in ${鈩�}^{3}$ . What is $m$ in this case?

(b) Show that any subspace $V$ of ${鈩�}^{n}$ can be represented as the image of a matrix.

In Exercises 25through 30, find the matrix Bof the linear transformation $T (\overset{鈫�}{x}) = A (\overset{鈫�}{x})$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .

$A = (\begin{matrix} 0 & 1 \\ 2 & 3 \end{matrix}); {\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 1 \end{matrix}]$

In Exercises 25 through 30, find the matrixBof the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .

$A = [\begin{matrix} 4 & 2 & - 4 \\ 2 & 1 & - 2 \\ - 4 & - 2 & 4 \end{matrix}]; {\overset{鈫�}{v}}_{1} = [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 0 \\ 2 \\ 1 \end{matrix}], {\overset{鈫�}{v}}_{3} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]$

Explain why you need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. See Theorem 3.3.4b.

Consider a linear transformation T from ${鈩�}^{n}$ to ${鈩�}^{p}$ and some linearly independent vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{m}$ in ${鈩�}^{n}$ . Are the vectors $T ({\overset{鈫�}{v}}_{1}), T ({\overset{鈫�}{v}}_{2}), . . . T ({\overset{鈫�}{v}}_{m})$ necessarily linearly independent? How can ${鈩�}^{p}$ you tell?

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