Exercises 4.8 Exercises
Exercise Group.
In the following exercises, compute the matrix of a given linear transformation with respect to given bases.
2.
Let \(V=M_{1\times n}(F)\) and \(W=M_{n\times 1}(F)\) be vector spaces over a field \(F\text{.}\) Let \(e_i=(\delta_{i1},\delta_{i2},\ldots,\delta_{in})\text{.}\) Consider ordered bases \(\mathfrak{B}_V=\{e_i:1\leq i\leq n\}\) and \(\mathfrak{B}_W=(e_i^t:1\leq i\leq n)\) of \(V\) and \(W\text{,}\) respectively. Consider the linear transformation \(T\colon V\to W\) given by \(x\mapsto x^t\text{.}\)3.
Suppose that \(V,W\) are finite-dimensional vector spaces over a field \(F\text{.}\) Let \(\mathfrak{B}_V=(v_1,v_2,\ldots,v_n)\) and \(\mathfrak{B}_W=(w_1,w_2,\ldots,w_m)\) be ordered bases of \(V\) and \(W\text{,}\) respectively. Consider the following set.
\begin{equation*}
\mathfrak{B}_V\bigoplus\mathfrak{B}_W=((v_1,0),(v_2,0),\ldots,(v_n,0),(0,w_1),(0,w_2),\ldots,(0,w_m))
\end{equation*}
This is an ordered basis of \(V\bigoplus W\text{.}\) Consider the projection \(\pi\colon V\bigoplus W\to V\) given by \((v,w)\mapsto v\text{.}\)4.
Let \(\mathfrak{B}_1=(e_1,e_2)\) be the standard ordered basis of \(\R^2\) (refer Example 3.6.1) and let \(\mathfrak{B}_2=(e_1+e_2,e_1-e_2)\) be another ordered basis of \(\R^2\) over \(\R\text{.}\) Let \(T_1\colon\R^2\to\R^2\) be defined by
\begin{equation*}
\alpha e_1+\beta e_2\mapsto\alpha(e_1+e_2)+\beta(e_1-e_2)\text{.}
\end{equation*}
Let \(T_2\colon\R^2\to\R^2\) be defined by
\begin{equation*}
\gamma_1(e_1+e_2)+\gamma_2(e_1-e_2)\mapsto\gamma_1e_1+\gamma_2e_2\text{.}
\end{equation*}
Find \([T_1]_{\mathfrak{B}_1}^{\mathfrak{B}_1}\) and \([T_2]_{\mathfrak{B}_2}^{\mathfrak{B}_1}\text{.}\)
