Chapter Test B

2. Hint: First write $v_{3}$ $v_{3}$ as a linear combination of $v_{1}$ $v_{1}$ and $v_{2}$ $v_{2}$ .
4. Hint: The range of L is a 2-dimensional subspace of $R^{3}$ $R^{3}$ . Find the two basis vectors.
5. Hint: To find the matrix representation A determine the effect of L on $e_{1}$ $e_{1}$ and $e_{2}$ $e_{2}$ .
6. Hint: To find the matrix representation A determine the effect of L on $e_{1}$ $e_{1}$ and $e_{2}$ $e_{2}$ .
8. Hint: The standard matrix representation for L is

$A = [\begin{matrix} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}]$ $A = [\begin{matrix} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}]$

Use $U = (u_{1}, u_{2})$ $U = (u_{1}, u_{2})$ and A to determine the matrix representation of L with respect to ${u_{1}, u_{2}}$ ${u_{1}, u_{2}}$
9.
1. (b) Hint: $I = S^{- 1} I S .$ $I = S^{- 1} I S .$

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Table of Contents for Chapter Test B