Define the linear transformation T: Rn→Rm by T(v) = Av. Use the matrix A to (a) determine the dimensions of Rn and Rm, (b) find the image of v, and (c) find the preimage of w. 1ál v = (6, 1, 1), w = (3, 5)A =%3D2

Let T be a linear transformation T: Rn→Rm such that T(v)=Av. A=012-200, v=6,1,1, w=3,5

Answered: 1 ál v = (6, 1, 1), w = (3, 5) A = %3D…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Define the linear transformation T: Rⁿ→R^m by T(v) = Av. Use the matrix A to (a) determine the dimensions of Rⁿ and R^m, (b) find the image of v, and (c) find the preimage of w.

1
ál v = (6, 1, 1), w = (3, 5)
A =
%3D
2

Expert Solution

Step 1 Given:

Let T be a linear transformation $T : R^{n} \to R^{m}$ such that $T (v) = A v$ .

$A = [\begin{matrix} 0 & 1 & 2 \\ - 2 & 0 & 0 \end{matrix}]$ , $v = (6, 1, 1)$ , $w = (3, 5)$

Step 2 (a)

To determine: The dimensions of $ℝ^{n}$ and $ℝ^{m}$ .

Here v shows the preimage.

The dimension of v is 3.

Thus, the dimension for $ℝ^{n}$ is 3 ( $ℝ^{3}$ )

Also, w is the image.

The dimension of w is 2.

Thus, the dimension for $ℝ^{m}$ is 2 ( $ℝ^{2}$ )

Step 3 (b)

To find: The image of v.

Let w be the image of v.

Then,

$A v = w$

$\begin{array}{rcl} w & = & [\begin{matrix} 0 & 1 & 2 \\ - 2 & 0 & 0 \end{matrix}] [\begin{matrix} 6 \\ 1 \\ 1 \end{matrix}] \\ = & [\begin{matrix} 0 \times 6 + 1 \times 1 + 2 \times 1 \\ - 2 \times 6 + 0 \times 1 + 0 \times 1 \end{matrix}] \\ = & [\begin{matrix} 0 + 1 + 2 \\ - 12 + 0 + 0 \end{matrix}] \\ = & [\begin{matrix} 3 \\ - 12 \end{matrix}] \end{array}$

The image of v is $(3, - 12)$ .

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