1.
\(T\colon M_{2\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}y\\-x\end{pmatrix}
\end{equation*}
Exercises 1.8 Exercises
A few exercises related to the concept of linear maps are listed below.
Exercise Group.
Consider the following maps. In each of the following cases do the following.
- Check if the map defined is linear or not.
- If the map is linear then find the matrix associated to it as outlined in Definition 1.7.5.
- Draw the images of standard basis vectors (see Definition 1.7.2).
- If the map is linear and if its associated matrix, \([T]=A\text{,}\) is invertible then find the linear map associated to \(A^{-1}\text{,}\) \(\ell_{A^{-1}}\) (see Remark 1.7.7). Compute \(T\circ\ell_{A^{-1}}\) and \(\ell_{A^{-1}}\circ T\text{.}\)
2.
\(T\colon M_{2\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\-y\end{pmatrix}
\end{equation*}
3.
\(T\colon M_{2\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x+2\\y\end{pmatrix}
\end{equation*}
4.
\(T\colon M_{2\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}\alpha\, x\\\beta \,y\end{pmatrix}
\end{equation*}
where, \(\alpha\) and \(\beta\) are real numbers.
5.
\(T\colon M_{2\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x+y\\x-y\end{pmatrix}
\end{equation*}
6.
\(T\colon M_{2\times 1}(\R)\to \R\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\end{pmatrix}=x
\end{equation*}
7.
\(T\colon M_{3\times 1}(\R)\to M_{2\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}y\\z\end{pmatrix}
\end{equation*}
8.
\(T\colon M_{3\times 1}(\R)\to M_{3\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}x+y\\y-z\\x+z\end{pmatrix}
\end{equation*}
9.
\(T\colon M_{3\times 1}(\R)\to M_{3\times 1}(\R)\) given by
\begin{equation*}
T\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}\cos x\\y\\z\end{pmatrix}
\end{equation*}
10.
Assume that \(T\colon M_{3\times 1}(\R)\to M_{3\times 1}(\R)\) is a linear map and
\begin{equation*}
T(e_1)=\begin{pmatrix}1\\1\\1\end{pmatrix},\quad T(e_2)=\begin{pmatrix}-1\\-1\\-1\end{pmatrix},\quad\text{and}\quad T(e_3)=\begin{pmatrix}1\\-1\\1\end{pmatrix}.
\end{equation*}
Find \(T\begin{pmatrix}2\\-3\\8\end{pmatrix}\text{.}\)
