Invertibility and Isomorphisms

Suppose $T \in \mathcal{L}(V, W)$ is invertible. Show that $T^{-1}$ is invertible and

Suppose $T \in \mathcal{L}(U, V)$ and $S \in \mathcal{L}(V, W)$ are both invertible linear maps. Prove that $S T \in \mathcal{L}(U, W)$ is invertible and that $(S T)^{-1} = T^{-1} S^{-1}$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that the following are equivalent.

1. $T$ is invertible.

2. $T v_{1}, \ldots , T v_{n}$ is a basis of $V$ for every basis $v_{1}, \ldots , v_{n}$ of $V$.

3. $T v_{1}, \ldots , T v_{n}$ is a basis of $V$ for some basis $v_{1}, \ldots , v_{n}$ of $V$.

Suppose $V$ is finite-dimensional and $\operatorname {dim} V > 1$. Prove that the set of noninvertible linear maps from $V$ to itself is not a subspace of $\mathcal{L}(V)$.

Suppose $V$ is finite-dimensional, $U$ is a subspace of $V$, and $S \in \mathcal{L}(U, V)$. Prove that there exists an invertible linear map $T$ from $V$ to itself such that $T u = S u$ for every $u \in U$ if and only if $S$ is injective.

Suppose that $W$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$. Prove that $\operatorname {null}S =\operatorname {null}T$ if and only if there exists an invertible $E \in \mathcal{L}(W)$ such that $S = E T$.

Suppose that $V$ is finite-dimensional and $S, T \in \mathcal{L}(V, W)$. Prove that range $S =$ range $T$ if and only if there exists an invertible $E \in \mathcal{L}(V)$ such that $S = T E$.

Suppose $V$ and $W$ are finite-dimensional and $S, T \in \mathcal{L}(V, W)$. Prove that there exist invertible $E_{1} \in \mathcal{L}(V)$ and $E_{2} \in \mathcal{L}(W)$ such that $S = E_{2} T E_{1}$ if and only if $\operatorname {dim}\operatorname {null}S = \operatorname {dim}\operatorname {null}T$.

Suppose $V$ is finite-dimensional and $T: V \rightarrow W$ is a surjective linear map of $V$ onto $W$. Prove that there is a subspace $U$ of $V$ such that $\left.T\right|_{U}$ is an isomorphism of $U$ onto $W$.

Suppose $V$ and $W$ are finite-dimensional and $U$ is a subspace of $V$. Let

Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$. Prove that

Suppose $V$ is finite-dimensional and $S, T, U \in \mathcal{L}(V)$ and $S T U = I$. Show that $T$ is invertible and that $T^{-1} = U S$.

Show that the result in Exercise 12 can fail without the hypothesis that $V$ is finite-dimensional.

Prove or give a counterexample: If $V$ is a finite-dimensional vector space and $R, S, T \in \mathcal{L}(V)$ are such that $R S T$ is surjective, then $S$ is injective.

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{m}$ is a list in $V$ such that $T v_{1}, \ldots , T v_{m}$ spans $V$. Prove that $v_{1}, \ldots , v_{m}$ spans $V$.

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

Suppose $V$ is finite-dimensional and $S \in \mathcal{L}(V)$. Define $\mathcal{A} \in \mathcal{L}(\mathcal{L}(V))$ by

for $T \in \mathcal{L}(V)$.

Show that $V$ and $\mathcal{L}(\mathbf{F}, V)$ are isomorphic vector spaces.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ has the same matrix with respect to every basis of $V$ if and only if $T$ is a scalar multiple of the identity operator.

Suppose $q \in \mathcal{P}(\mathbb {R})$. Prove that there exists a polynomial $p \in \mathcal{P}(\mathbb {R})$ such that

for all $x \in \mathbb {R}$.

Suppose $n$ is a positive integer and $A_{j, k} \in \mathbf{F}$ for all $j, k = 1, \ldots , n$. Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables).

1. The trivial solution $x_{1} = \cdots = x_{n} = 0$ is the only solution to the homogeneous system of equations

   $$\begin{aligned} & \sum _{k = 1}^{n} A_{1, k} x_{k} = 0 \\ & \vdots \\ & \sum _{k = 1}^{n} A_{n, k} x_{k} = 0 \end{aligned}$$

2. For every $c_{1}, \ldots , c_{n} \in \mathbf{F}$, there exists a solution to the system of equations

   $$\begin{gathered} \sum _{k = 1}^{n} A_{1, k} x_{k} = c_{1} \\ \vdots \\ \sum _{k = 1}^{n} A_{n, k} x_{k} = c_{n} \end{gathered}$$

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$. Prove that

Suppose that $u_{1}, \ldots , u_{n}$ and $v_{1}, \ldots , v_{n}$ are bases of $V$. Let $T \in \mathcal{L}(V)$ be such that $T v_{k} = u_{k}$ for each $k = 1, \ldots , n$. Prove that

Suppose $A$ and $B$ are square matrices of the same size and $A B = I$. Prove that $B A = I$.