Q27E In Exercises 25 聽through聽 30, ... [FREE SOLUTION]

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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: Q27E (page 160)

In Exercises 25 through 30, find the matrixBof the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .
$A = [\begin{matrix} 4 & 2 & - 4 \\ 2 & 1 & - 2 \\ - 4 & - 2 & 4 \end{matrix}]; {\overset{鈫�}{v}}_{1} = [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 0 \\ 2 \\ 1 \end{matrix}], {\overset{鈫�}{v}}_{3} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]$

Short Answer

Expert verified

The matrix is, $B = [\begin{matrix} 9 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ .

Step by step solution

Consider the vectors.

The matrix and vectors are,

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

Compute the matrix using formula.

The formula is, $B = S^{- 1} A S$ .

Compute the matrix and the inverse of S .

$S = ({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}) = [\begin{matrix} 2 & 0 & 1 \\ 1 & 2 & 0 \\ - 2 & 1 & 1 \end{matrix}] 鈬� S^{- 1} = \frac{1}{9} [\begin{matrix} 2 & 1 & - 2 \\ - 1 & 4 & 1 \\ 5 & - 2 & 4 \end{matrix}]$

Substitute these values in the formula.

$B = S^{- 1} A S 鈬� B = \frac{1}{9} [\begin{matrix} 2 & 1 & - 2 \\ - 1 & 4 & 1 \\ 5 & - 2 & 4 \end{matrix}] [\begin{matrix} 4 & 2 & - 4 \\ 2 & 1 & - 2 \\ - 4 & - 2 & 4 \end{matrix}] [\begin{matrix} 2 & 0 & 1 \\ 1 & 2 & 0 \\ - 2 & 1 & 1 \end{matrix}] 鈭� B = [\begin{matrix} 9 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

Final answer.

The matrix is, $B = [\begin{matrix} 9 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ .

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Short Answer

Step by step solution

Consider the vectors.

Compute the matrix using formula.

Final answer.

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91影视

In Exercises 25 through 30, find the matrixBof the linear transformationT(x鈫�)=Ax鈫� with respect to the basis J=(v鈫�1,鈥�..,v鈫�m).A=[42-421-2-4-24];v鈫�1=[21-2],v鈫�2=[021],v鈫�3=[101]

Short Answer

Step by step solution

Consider the vectors.

Compute the matrix using formula.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Probability and Statistics

Statistics

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.