Null Spaces and Ranges

Give an example of a linear map $T$ with $\operatorname {dim}\operatorname {null}T = 3$ and $\operatorname {dim}$ range $T = 2$.

Suppose $S, T \in \mathcal{L}(V)$ are such that range $S \subseteq \operatorname {null}T$. Prove that $(S T)^{2} = 0$.

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. Define $T \in \mathcal{L}\left(\mathbf{F}^{m}, V\right)$ by

Show that $\left\{ T \in \mathcal{L}\left(\mathbb {R}^{5}, \mathbb {R}^{4}\right) \text{ : dim } \operatorname {null} T > 2\right\}$ is not a subspace of $\mathcal{L}\left(\mathbb {R}^{5}, \mathbb {R}^{4}\right)$.

Give an example of $T \in \mathcal{L}\left(\mathbb {R}^{4}\right)$ such that range $T =\operatorname {null}T$.

Prove that there does not exist $T \in \mathcal{L}\left(\mathbb {R}^{5}\right)$ such that range $T =\operatorname {null}T$.

Suppose $V$ and $W$ are finite-dimensional with $2 \leq \operatorname {dim} V \leq \operatorname {dim} W$. Show that $\left\{ T \in \mathcal{L}(V, W): T\text{ is not injective }\right\}$ is not a subspace of $\mathcal{L}(V, W)$.

Suppose $V$ and $W$ are finite-dimensional with $\operatorname {dim} V \geq \operatorname {dim} W \geq 2$. Show that $\left\{ T \in \mathcal{L}(V, W): T\text{ is not surjective }\right\}$ is not a subspace of $\mathcal{L}(V, W)$.

Suppose $T \in \mathcal{L}(V, W)$ is injective and $v_{1}, \ldots , v_{n}$ is linearly independent in $V$. Prove that $T v_{1}, \ldots , T v_{n}$ is linearly independent in $W$.

Suppose $v_{1}, \ldots , v_{n}$ spans $V$ and $T \in \mathcal{L}(V, W)$. Show that $T v_{1}, \ldots , T v_{n}$ spans range $T$.

Suppose that $V$ is finite-dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that

Suppose $T$ is a linear map from $\mathbf{F}^{4}$ to $\mathbf{F}^{2}$ such that

Prove that $T$ is surjective.

Suppose $U$ is a three-dimensional subspace of $\mathbb {R}^{8}$ and that $T$ is a linear map from $\mathbb {R}^{8}$ to $\mathbb {R}^{5}$ such that $\operatorname {null}T = U$. Prove that $T$ is surjective.

Prove that there does not exist a linear map from $\mathbf{F}^{5}$ to $\mathbf{F}^{2}$ whose null space equals $\left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \in \mathbf{F}^{5}: x_{1} = 3 x_{2} \text{ and } x_{3} = x_{4} = x_{5}\right\}$.

Suppose there exists a linear map on $V$ whose null space and range are both finite-dimensional. Prove that $V$ is finite-dimensional.

Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists an injective linear map from $V$ to $W$ if and only if $\operatorname {dim} V \leq \operatorname {dim} W$.

Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists a surjective linear map from $V$ onto $W$ if and only if $\operatorname {dim} V \geq \operatorname {dim} W$.

Suppose $V$ and $W$ are finite-dimensional and that $U$ is a subspace of $V$. Prove that there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname {null}T = U$ if and only if $\operatorname {dim} U \geq \operatorname {dim} V-\operatorname {dim} W$.

Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is injective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $S T$ is the identity operator on $V$.

Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is surjective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $T S$ is the identity operator on $W$.

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V, W)$, and $U$ is a subspace of $W$. Prove that $\left\{ v \in V: T v \in U\right\}$ is a subspace of $V$ and

Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$. Prove that

Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$. Prove that dim range $S T \leq \min \left\{ \operatorname {dim} \operatorname {range} S, \operatorname {dim} \operatorname {range} T\right\}$.

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

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

Suppose $P \in \mathcal{L}(V)$ and $P^{2} = P$. Prove that $V =\operatorname {null}P \oplus$ range $P$.

Suppose $D \in \mathcal{L}(\mathcal{P}(\mathbb {R}))$ is such that $\operatorname {deg} D p = (\operatorname {deg} p)-1$ for every nonconstant polynomial $p \in \mathcal{P}(\mathbb {R})$. Prove that $D$ is surjective.

Suppose $p \in \mathcal{P}(\mathbb {R})$. Prove that there exists a polynomial $q \in \mathcal{P}(\mathbb {R})$ such that $5 q^{\prime \prime }+3 q^{\prime } = p$.

Suppose $\varphi \in \mathcal{L}(V, \mathbf{F})$ and $\varphi \neq 0$. Suppose $u \in V$ is not in $\operatorname {null}\varphi$. Prove that

Suppose $V$ is finite-dimensional, $X$ is a subspace of $V$, and $Y$ is a finitedimensional subspace of $W$. Prove that there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname {null}T = X$ and range $T = Y$ if and only if $\operatorname {dim} X+\operatorname {dim} Y = \operatorname {dim} V$.

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 1$. Show that if $\varphi : \mathcal{L}(V) \rightarrow \mathbf{F}$ is a linear map such that $\varphi (S T) = \varphi (S) \varphi (T)$ for all $S, T \in \mathcal{L}(V)$, then $\varphi = 0$.

Suppose that $V$ and $W$ are real vector spaces and $T \in \mathcal{L}(V, W)$. Define $T_{\mathrm{C}}: V_{\mathrm{C}} \rightarrow W_{\mathrm{C}}$ by

for all $u, v \in V$. The linear map $T_{\mathrm{C}}$ is called the complexification of the linear map $T$.