Answered: 2 A = | 1 –1 1 B' = {(-1, 1, 0), (2, 1,…

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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Question

Let A be the matrix for T: R³→R³ relative to the standard basis. Find the diagonal matrix A′ for T relative to the basis B′.

2
A = | 1
–1
1
B' = {(-1, 1, 0), (2, 1, 0), (0, 0, 1)}

Expert Solution

Step 1

A square matrix is said to be a diagonal matrix if all its non-diagonal entries are zero except the

diagonal entries.

We have to write the diagonal matrix A' for the linear transformation T relative to the basis $B'$

using the given matrix A.

Step 2

We know that $T (x) = A x$ . We have $A = [\begin{matrix} 0 & 2 & 0 \\ 1 & - 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ .

Now,

$\begin{array}{rcl} T (- 1, 1, 0) & = & [\begin{matrix} 0 & 2 & 0 \\ 1 & - 1 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} - 1 \\ 1 \\ 0 \end{matrix}] \\ = & [\begin{matrix} 2 \\ - 2 \\ 0 \end{matrix}] \\ = & - 2 [\begin{matrix} - 1 \\ 1 \\ 0 \end{matrix}] \\ ∴ T (- 1, 1, 0) & = & - 2 [\begin{matrix} - 1 \\ 1 \\ 0 \end{matrix}] + 0 [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \end{array}$ .

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