Homework 13: Standard Matrices, Compositions

1. Section 1.3, #5(a), (b).
2. Section 4.10, #9, 11.
3. Find the standard matrix for each of the following linear transformations.
   1. \(S:\bR^2\to \bR^2\) given by rotating \(2\theta\) clockwise about the origin.
   2. \(T:\bR^2\to \bR^2\) given by reflecting across the line \(y=-x\).
   3. \(R:\bR^3\to \bR^3\) given by rotating \(180^{\circ}\) about the line passing through points \((-1,0,-1)\) and \((1,0,1)\).
4. Find the result of rotating the vector \(\vthree{2}{4}{7}\) by 180\(^{\circ}\) about the line through the points \((-1,0,-1)\) and \((1,0,1)\).
5. In \(\bR^3\), let \(T\) be counterclockwise rotation by \(\theta\) about the \(z\)-axis, and let \(S\) be counterclockwise rotation by \(\psi\) about the \(y\)-axis. (Here counterclockwise means as viewed from the positive axis.) Let \(R\) be counterclockwise rotation by \(\theta\) about the \(z\)-axis followed by counterclockwise rotation by \(\psi\) in \(\bR^3\) about the \(y\)-axis. Find standard matrices for \(T\), \(S\), and \(R\).
   Hint: The standard matrix for \(R\) is obtained by multiplying the other two matrices (in the correct order).
6. In this problem you will prove the following trigonometric identities: $$ \begin{align} \cos(x+y)&=\cos x \cos y- \sin x \sin y\\ \sin(x+y)&= \sin x \cos y + \cos x \sin y \end{align} $$ Use the following strategy. Let \(T\) be counterclockwise rotation by \(x\) and \(S\) counterclockwise rotation by \(y\) (both in \(\bR^2\)). Find the standard matrix for \(S\circ T\) in two ways: (i) by multiplying the standard matrices for \(S\) and \(T\), and (ii) by observing that \(S\circ T\) is rotation by \(x+y\). The matrices obtained in (i) and (ii) must be equal, so their entries must be equal.
7. Use the trigonometric identities in Problem 6 to find formulas for \(\cos (2x)\) and \(\sin (2x)\).
8. Suppose that there exists a linear transformation from \(V\) onto \(W\), where both \(V\) and \(W\) are finite-dimensional vector spaces. Is it possible for the dimension of \(W\) to be greater than the dimension of \(V\)? Justify your answer.
9. Give an isomorphism \(T:M_{2,3}\to \bR^6\). No justification required.
10. Give an example of a transformation \(T:\bR^2\to \bR^2\) which is bijective but not an isomorphism.