Homework 14: The Matrix of a Linear Transformation

1. Section 1.5, #9, 11(a), 16, 20(a).
2. Section 8.4, #8.
3. Construct the following:
   1. A matrix \(A\) such that the matrix transformation \(T(\bx)=A\bx\) is one-to-one but not onto.
   2. A matrix \(B\) such that the matrix transformation \(T(\bx)=B\bx\) is onto but not one-to-one.
   3. A matrix \(C\) such that the matrix transformation \(T(\bx)=C\bx\) is one-to-one and onto.
   Briefly explain why your given matrices satify the required properties.
4. Let \(T:V\to W\) be linear, and let \(\cB=\{\bu_1,\bu_2\}\) be a basis for \(V\) and \(\cB'=\{\bw_1,\bw_2,\bw_3\}\) a basis for \(W\). Suppose that \(T(\bu_1)=3\bw_1+5\bw_2-7\bw_3\) and \(T(\bu_2)=2\bw_1+4\bw_3\).
   1. Find \([T]_{\cB',\cB}\).
   2. If \([\bx]_\cB=\vtwo{\pe 7}{-2}\), then what is \([T(\bx)]_{\cB'}\)?
5. Let \(\cB\) be the standard basis of \(\bR^2\), let \(\cC\) be the basis of \(\bR^3\) consisting of the three vectors \(\bu_1=\vthree{1}{1}{0}, \bu_2=\vthree{1}{0}{1},\bu_3=\vthree{0}{1}{1}\), and let \(\cD\) be the standard basis of \(\bR^3\). Let \(T:\bR^2\to \bR^3\) be the linear transformation defined by \(T\left(\vtwo{a}{b}\right)=a\bu_1+b\bu_2+(a+b)\bu_3\). Find \([T]_{\cC,\cB}\) and \([T]_{\cD,\cB}\).
6. Let \(T:\bR^2\to \bR^2\) be reflection across the line \(y=\frac{1}{3}x\). Let \(\cB=\left\{\vtwo{3}{1},\vtwo{-1}{\pe 3}\right\}\), a basis of \(\bR^2\). Find \([T]_{\cB,\cB}\).