Q29 E In Exercises 聽25 聽through 30 ,... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q29 E (page 160)

In Exercises 25 through 30 , find the matrix B of the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .
$A = [\begin{matrix} - 1 & 1 & 0 \\ 0 & - 2 & 2 \\ 3 & - 9 & 6 \end{matrix}]; {\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]; {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]; {\overset{鈫�}{v}}_{3} = [\begin{matrix} 1 \\ 3 \\ 6 \end{matrix}]$

Short Answer

Expert verified

The matrix is, $B = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}]$ .

Step by step solution

Consider the vectors.

The matrix and vectors are,

$A = [\begin{matrix} - 1 & 1 & 0 \\ 0 & - 2 & 2 \\ 3 & - 9 & 6 \end{matrix}]; {\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]; {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]; {\overset{鈫�}{v}}_{3} = [\begin{matrix} 1 \\ 3 \\ 6 \end{matrix}]$

Compute the matrix using formula.

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

Compute the matrix and the inverse of S.

$S = ({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}) = [\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{matrix}] 鈬� S^{- 1} = \frac{1}{1} [\begin{matrix} 3 & - 3 & 1 \\ - 3 & 5 & - 2 \\ 1 & - 2 & 1 \end{matrix}]$

Substitute these values in the formula.

$B = S^{- 1} A S 鈬� B = \frac{1}{1} [\begin{matrix} 3 & - 3 & 1 \\ - 3 & 5 & - 2 \\ 1 & - 2 & 1 \end{matrix}] [\begin{matrix} - 1 & 1 & 0 \\ 0 & - 2 & 2 \\ 3 & - 9 & 6 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{matrix}] 鈭� B = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}]$

Final answer.

The matrix is, $B = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}]$ .

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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 matrix B of the linear transformation T(x鈫�)=Ax鈫� with respect to the basisJ=(v鈫�1,鈥�..,v鈫�m).A=[-1100-223-96];v鈫�1=[111];v鈫�2=[123;v鈫�3=136]

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

Geometry

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.