Chapter 7

Show that the following are equivalent for a linear transformation \(T: V \rightarrow W\). 1\. ker \(T=V\) 2\. \(\operatorname{im} T=\\{\mathbf{0}\\}\) 3\. \(T=0\)

Let \(T: V \rightarrow W\) be a linear transformation. a. If \(T\) is one-to-one and \(T R=T R_{1}\) for transformations \(R\) and \(R_{1}: U \rightarrow V\), show that \(R=R_{1}\). b. If \(T\) is onto and \(S T=S_{1} T\) for transformations \(S\) and \(S_{1}: W \rightarrow U,\) show that \(S=S_{1}\).

Let \(S: V \rightarrow W\) and \(T: V \rightarrow W\) be linear transformations. Given \(a\) in \(\mathbb{R},\) define functions \((S+T): V \rightarrow W\) and \((a T): V \rightarrow W\) by \((S+T)(\mathbf{v})=\) \(S(\mathbf{v})+T(\mathbf{v})\) and \((a T)(\mathbf{v})=a T(\mathbf{v})\) for all \(\mathbf{v}\) in \(V\). Show that \(S+T\) and \(a T\) are linear transformations.

Describe all linear transformations \(T: \mathbb{R} \rightarrow V\).

Consider the linear transformations \(V \stackrel{T}{\rightarrow} W \stackrel{R}{\rightarrow} U\). a. Show that ker \(T \subseteq \operatorname{ker} R T\). b. Show that \(\operatorname{im} R T \subseteq \operatorname{im} R\).

Let \(A\) be an \(m \times n\) matrix of \(\operatorname{rank} r\) Thinking of \(\mathbb{R}^{n}\) as rows, define \(V=\left\\{\mathbf{x}\right.\) in \(\left.\mathbb{R}^{m} \mid \mathbf{x} A=\mathbf{0}\right\\}\). Show that \(\operatorname{dim} V=m-r\).

Let \(V \stackrel{T}{\rightarrow} U \stackrel{S}{\rightarrow} W\) be linear transformations. a. If \(S T\) is one-to-one, show that \(T\) is one-to-one and that \(\operatorname{dim} V \leq \operatorname{dim} U\) b. If \(S T\) is onto, show that \(S\) is onto and that \(\operatorname{dim} W \leq \operatorname{dim} U\)

Consider $$ V=\left\\{\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \mid a+c=b+d\right\\} $$ a. Consider \(S: \mathbf{M}_{22} \rightarrow \mathbb{R}\) with \(S\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]=a+c-\) \(b-d\). Show that \(S\) is linear and onto and that \(V\) is a subspace of \(\mathbf{M}_{22}\). Compute \(\operatorname{dim} V\). b. Consider \(T: V \rightarrow \mathbb{R}\) with \(T\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]=a+c\). Show that \(T\) is linear and onto, and use this information to compute \(\operatorname{dim}(\) ker \(T)\).

Let \(T: V \rightarrow V\) be a linear transformation. Show that \(T^{2}=1_{V}\) if and only if \(T\) is invertible and \(T=T^{-1}\)

Define \(T: \mathbf{P}_{n} \rightarrow \mathbb{R}\) by \(T[p(x)]=\) the sum of all the coefficients of \(p(x)\). a. Use the dimension theorem to show that \(\operatorname{dim}(\operatorname{ker} T)=n\) b. Conclude that \(\left\\{x-1, x^{2}-1, \ldots, x^{n}-1\right\\}\) is a basis of \(\operatorname{ker} T\)