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Give 2 2 matrices A so that for any x R2 we have, respectively: a. Ax is the vector
Chapter 2, Problem 5(choose chapter or problem)
Give \(2 \times 2\) matrices A so that for any \(\mathbf{x} \in \mathbb{R}^{2}\) we have, respectively:
a. Ax is the vector whose components are, respectively, the sum and difference of the components of x.
b. Ax is the vector obtained by projecting x onto the line \(x_{1}=x_{2} \text { in } \mathbb{R}^{2}\).
c. Ax is the vector obtained by first reflecting x across the line \(x_{1}=0\) and then reflecting the resulting vector across the line \(x_{2}=x_{1}\).
d. Ax is the vector obtained by projecting x onto the line \(2 x_{1}-x_{2}=0\).
e. Ax is the vector obtained by first projecting x onto the line \(2 x_{1}-x_{2}=0\) and then rotating the resulting vector \(\pi / 2\) counterclockwise.
f. Ax is the vector obtained by first rotating x an angle of \(\pi / 2\) counterclockwise and then projecting the resulting vector onto the line \(2 x_{1}-x_{2}=0\).
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QUESTION:
Give \(2 \times 2\) matrices A so that for any \(\mathbf{x} \in \mathbb{R}^{2}\) we have, respectively:
a. Ax is the vector whose components are, respectively, the sum and difference of the components of x.
b. Ax is the vector obtained by projecting x onto the line \(x_{1}=x_{2} \text { in } \mathbb{R}^{2}\).
c. Ax is the vector obtained by first reflecting x across the line \(x_{1}=0\) and then reflecting the resulting vector across the line \(x_{2}=x_{1}\).
d. Ax is the vector obtained by projecting x onto the line \(2 x_{1}-x_{2}=0\).
e. Ax is the vector obtained by first projecting x onto the line \(2 x_{1}-x_{2}=0\) and then rotating the resulting vector \(\pi / 2\) counterclockwise.
f. Ax is the vector obtained by first rotating x an angle of \(\pi / 2\) counterclockwise and then projecting the resulting vector onto the line \(2 x_{1}-x_{2}=0\).
ANSWER:Step 1 of 7
A is a 2x2 matrix and \(x \in {R^2}\).
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Review this written solution for 964540) viewed: 334 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 2.2 - Problem 5
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