Q31E Find the basis of all聽2脳2 matr... [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 4: Q31E (page 176)

Find the basis of all $2 脳 2$ matrixA such that $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ , and determine its dimension.

Short Answer

Expert verified

The dimension ofS a 2 basis of is which is spanned by $S p a n \{[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}]\}$ .

Step by step solution

Determine the matrix .

Consider the equation $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ where $S = [\begin{matrix} a & b \\ c & d \end{matrix}]$ .

Substitute the value $[\begin{matrix} a & b \\ c & d \end{matrix}]$ forS in the equation $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ as follows.

$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} c & d \\ a & b \end{matrix}] = [\begin{matrix} a & - b \\ c & - d \end{matrix}]$

Compare the equation both side as follows.

$\begin{array}{l} c = a \\ d = - b \end{array}$

Substitute the values a for c and -b for d in the equation $S = [\begin{matrix} a & b \\ c & d \end{matrix}]$ as follows.

$\begin{array}{l} S = [\begin{matrix} a & b \\ c & d \end{matrix}] \\ S = [\begin{matrix} a & b \\ a & - b \end{matrix}] \end{array}$

Therefore, any matrix in the form $S = [\begin{matrix} a & b \\ a & - b \end{matrix}]$ satisfy the equation $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ .

Determine the basis of the matrix .

Simplify the equation $S = [\begin{matrix} a & b \\ a & - b \end{matrix}]$ as follows.

$S = [\begin{matrix} a & b \\ a & - b \end{matrix}] S = [\begin{matrix} a & 0 \\ a & 0 \end{matrix}] + [\begin{matrix} 0 & b \\ 0 & - b \end{matrix}] S = a [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}] + b [\begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}]$

where $[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$ and $[\begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}]$ are linear independent.

Therefore, the matrixS is spanned by $S p a n \{[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}]\}$ .

Hence, the dimension of Sis 2 and spanned by $S p a n \{[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & - 1 \end{matrix}]\}$ .

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91影视

Find the basis of all $2 脳 2$ matrixA such that $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ , and determine its dimension.

Short Answer

Step by step solution

Determine the matrix .

Determine the basis of the matrix .

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91影视

Find the basis of all2脳2 matrixA such that[0110]S=S[100-1], and determine its dimension.

Short Answer

Step by step solution

Determine the matrix .

Determine the basis of the matrix .

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Find the basis of all $2 脳 2$ matrixA such that $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] S = S [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ , and determine its dimension.