Q3.4-21E In Exercises 9聽聽through 24, fi... [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 1: Q3.4-21E (page 1)

In Exercises 9through 24, find the matrix Bof the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ with respect to the basis $I = (\overset{鈬赌}{v_{1}}, \overset{鈬赌}{v_{2}})$ . For practice, solve each problem in three ways: (a) Use the formula $B = S^{- 1} A S$ , (b) use a commutative diagram (as in Examples 3 and 4 ), and (c)construct 鈥渃olumn by column.鈥�
role="math" localid="1664342971664" $A = (\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}); \overset{鈬赌}{v_{1}} = [\begin{matrix} 1 \\ 3 \end{matrix}]; \overset{鈬赌}{v_{2}} = [\begin{matrix} - 2 \\ 1 \end{matrix}]$

Short Answer

Expert verified

(a) The matrix is, $B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$ .

(b) The matrix is, $B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$ .

Step by step solution

Consider the vectors.

The vectors are,

$A = (\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}); \overset{鈬赌}{v_{1}} = [\begin{matrix} 1 \\ 3 \end{matrix}]; \overset{鈬赌}{v_{2}} = [\begin{matrix} - 2 \\ 1 \end{matrix}]$

Compute the matrix using the formula

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

Compute the matrix $S$ .

The inverse of the matrix is, $S = (\begin{matrix} a & b \\ c & d \end{matrix}) 鈬� S^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$

$S = (\overset{鈬赌}{v_{1}}, \overset{鈬赌}{v_{1}}) = (\begin{matrix} 1 & - 2 \\ 3 & 1 \end{matrix}) 鈬� S^{- 1} = \frac{1}{7} (\begin{matrix} 1 & 2 \\ - 3 & 1 \end{matrix})$

Substitute these values in the formula.

$B = S^{- 1} A S 鈬� B = \frac{1}{7} (\begin{matrix} 1 & 2 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}) (\begin{matrix} 1 & - 2 \\ 3 & 1 \end{matrix}) B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$

Compute the matrix using a commutative diagram.

The matrix is,

{(\begin{matrix} c_{1} \\ c_{2} \end{matrix})}_{I} = (\begin{matrix} a & b \\ c & d \end{matrix}) {(\begin{matrix} c_{1} \\ c_{2} \end{matrix})}_{I} = (\begin{matrix} a c_{1} & b c_{2} \\ c c_{1} & d c_{2} \end{matrix})

$a = 7, b = 0, c = 0, d = 0$

Substitute these values in the formula.

$B = (\begin{matrix} a & b \\ c & d \end{matrix}) 鈭� B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$

Compute the matrix by constructing columns.

The formula is, $T (\overset{鈬赌}{v_{1}}) = A \overset{鈬赌}{v_{1}}, T (\overset{鈬赌}{v_{2}}) = A \overset{鈬赌}{v_{2}}$

Compute the matrix

$T (\overset{鈬赌}{v_{1}}) = A \overset{鈬赌}{v_{1}} (\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}) (\begin{matrix} 1 \\ 3 \end{matrix}) = (\begin{matrix} 7 \\ 0 \end{matrix}) = \overset{鈬赌}{v_{1}} T (\overset{鈬赌}{v_{2}}) = A \overset{鈬赌}{v_{2}} (\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}) (\begin{matrix} - 2 \\ 1 \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) = - \overset{鈬赌}{v_{2}}$

Substitute these values in the formula.

$T (\overset{鈬赌}{v_{1}}) = (\begin{matrix} 7 \\ 0 \end{matrix}), T (\overset{鈬赌}{v_{2}}) = (\begin{matrix} 0 \\ 0 \end{matrix}) 鈭� B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$

Final answer.

(a) The matrix is, $B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$ .

(b) The matrix is, $B = (\begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix})$ .

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91影视

Short Answer

Step by step solution

Consider the vectors.

Compute the matrix using the formula

Compute the matrix using a commutative diagram.

Compute the matrix by constructing columns.

Final answer.

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