Homework 15: The Change of Basis Formula

1. Section 8.5, #10, 13.
2. Section 6.1, #34, 38.
3. Let \(T:\bR^2\to \bR^2\) be projection to the line \(y=\frac{1}{5}x\). Let \(\cB=\left\{\vtwo{5}{1},\vtwo{-1}{\pe 5}\right\}\) and \(\cC=\{\mathbf{e}_1,\mathbf{e}_2\}\), two bases of \(\bR^2\).
   1. Find \([T]_{\cB}\).
   2. Use (a) and the change of basis formula to find \([T]_{\cC}\), the standard matrix for \(T\).
   3. Find the projection of \(\vtwo{3}{7}\) to the line \(y=\frac{1}{5}x\), i.e., find \(T\left(\vtwo{3}{7}\right)\).
4. Let \(T:P_1\to P_1\) be the linear transformation given by \(T(p(x))= xp'(x) + p(x)\). Let \(\cB=\{1,x\}\) and \(\cC=\{1+2x,3+5x\}\), two bases for \(P_1\).
   1. Find the change of coordinates matrices \([I]_{\cB,\cC}\) and \([I]_{\cC,\cB}\).
   2. Find \([T]_{\cB}\).
   3. Use the change of basis formula and your answers to parts (a) and (b) to find \([T]_{\cC}\).
5. Let \(a_1,a_2,a_3\) be constants. Let \(T:\bR^3\to \bR^3\) be the linear operator defined by the formula $$ T \left( \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} \right) = \begin{bmatrix} a_1x_1\\ a_2x_2\\ a_3x_3 \end{bmatrix}$$
   1. Under what conditions will \(T\) have an inverse?
   2. Assuming the conditions determined in part (a) are satisfied, find a formula for \(T^{-1} \left( \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} \right)\).
6. Let \(\bu=\vtwo{u_1}{u_2}\) and \(\bv=\vtwo{v_1}{v_2}\). Show that \(\ip{\bu}{\bv} = 3u_1v_1\) does not define an inner product on \(\bR^2\). List all inner product axioms that fail to hold. For each axiom which does not hold, give a counterexample demonstrating this.