Exercises 5.7 Exercises
2.
Let \(\mathcal{P}_{n-1}(F)\) be the vector space of polynomials in one variable over a field \(F\) of degree at most \(n-1\text{.}\) Let \(a_1,a_2,\ldots,a_n\in F\) be distinct elements. Show that the following map is an \(F\)-isomorphism.
\begin{equation*}
T\colon\mathcal{P}_{n-1}(F)\longrightarrow F^n\quad\text{given by}\quad p\mapsto (p(a_1),\ldots,p(a_n)).
\end{equation*}
3.
Find all non-isomorphic subspaces of \(\R^2\text{.}\)4.
Let \(V\) be a vector space over a field \(F\text{.}\) Show that \(\Hom_F(F,V)\simeq V\text{.}\)5.
Let \(V, W\) be vector spaces over a field \(F\text{.}\) Show that \((V\bigoplus W)^*\simeq V^*\bigoplus W^*\text{.}\) (The generalization of this exercise can be found Example 6.4.1).6.
Let \(A\in M_{m\times n}(F)\) and \(L_A\colon F^n\to F^m\) be an \(F\)-linear map given by \(v\mapsto Av^t\text{.}\) Show that- \(L_A\) is injective if and only if \(rk(A)=n\text{.}\)
- \(L_A\) is surjective if and only if \(rk(A)=m\text{.}\)
- \(L_A\) is invertible if and only if \(n=m\) and \(A\) is invertible.
7.
Suppose that \(A\in{\rm GL}_n(F)\text{.}\) Consider the following map.
\begin{equation*}
T\colon F^n\to F^n\quad\text{given by}\quad (\alpha_1,\ldots,\alpha_n)\mapsto \tfrac{1}{\det(A)}\left(\det(A_1),\ldots,\det(A_n)\right),
\end{equation*}
where \(A_i\) are obtained by replacing the \(i\)-th column of \(A\) by \((\alpha_1,\ldots,\alpha_n)^t\text{.}\)
Show that the map defined above is a linear isomorphism. This will prove the Cramer’s rule 1 .
8.
Let \(V\) be a finite-dimensional vector space over a field \(F\text{.}\) Show that \(v_1,v_2,\ldots,v_r\in V\) are linearly independent over \(F\) if and only if there are \(f_1,f_2,\ldots,f_r\in V^*\) such that \(\det(f_i(v_j))\neq 0\text{.}\)Exercise Group.
In the following exercises, find bases so that the matrix representation of the given linear map has the following form.
\begin{equation*}
\begin{pmatrix}I_r\amp 0\\0\amp 0\end{pmatrix}
\end{equation*}
9.
\(T\colon\R^2\to\R\) given by \((x,y)\mapsto x\text{.}\)10.
\(tr\colon M_n(F)\to F\) given by \((a_{ij})\mapsto \sum_i a_{ii}\text{.}\)11.
\(T\colon M_{n}(F)\to M_n(F)\) given by
\begin{equation*}
\sum_{1\leq i,j\leq n} a_{ij}E_{ij}\mapsto\sum_{i\geq j}a_{ij}E_{ij}\text{.}
\end{equation*}
12.
\(T\colon M_n(F)\to M_n(F)\) given by \(A\mapsto E_{11}A\text{.}\)13.
\(\mathcal{D}_n\colon\mathcal{P}_n(F)\to\mathcal{P}_{n-1}(F)\) given by \(\sum_{i=0}^na_iX^i\mapsto\sum_{i=1}^{n-1}ia_iX^{i-1}\text{.}\)en.wikipedia.org/wiki/Cramer%27s_rule
