91Ó°ÊÓ

Prove that every linear map from Mat( \(n, 1, \mathbf{F}\) ) to Mat( \(m, 1, \mathbf{F}\) ) is given by a matrix multiplication. In other words, prove that if \(T \in \mathcal{L}(\operatorname{Mat}(n, 1, \mathbf{F}), \operatorname{Mat}(m, 1, \mathbf{F})),\) then there exists an \(m\) -by- \(n\) matrix \(A\) such that \(T B=A B\) for every \(B \in \operatorname{Mat}(n, 1, \mathbf{F})\)

Suppose that \(V\) is finite dimensional and \(S, T \in \mathcal{L}(V)\). Prove that \(S T\) is invertible if and only if both \(S\) and \(T\) are invertible.

Suppose that \(V\) is finite dimensional and \(S, T \in \mathcal{L}(V)\). Prove that \(S T=I\) if and only if \(T S=I\)

Suppose that \(V\) is finite dimensional and \(T \in \mathcal{L}(V) .\) Prove that \(T\) is a scalar multiple of the identity if and only if \(S T=T S\) for every \(S \in \mathcal{L}(V)\)

Prove that if \(V\) is finite dimensional with \(\operatorname{dim} V>1\), then the set of noninvertible operators on \(V\) is not a subspace of \(\mathcal{L}(V)\)

Suppose \(n\) is a positive integer and \(a_{i, j} \in \mathbf{F}\) for \(i, j=1, \ldots, n\) Prove that the following are equivalent: (a) The trivial solution \(x_{1}=\cdots=x_{n}=0\) is the only solution to the homogeneous system of equations $$\begin{array}{c}\sum_{k=1}^{n} a_{1, k} x_{k}=0 \\\\\vdots \\\\\sum_{k=1}^{n} a_{n, k} x_{k}=0\end{array}$$ (b) For every \(c_{1}, \ldots, c_{n} \in \mathbf{F},\) there exists a solution to the sys tem of equations $$\begin{array}{c}\sum_{k=1}^{n} a_{1, k} x_{k}=c_{1} \\ \vdots \\\\\sum_{k=1}^{n} a_{n, k} x_{k}=c_{n}\end{array}$$ Note that here we have the same number of equations as variables.