Vector Space of Linear Maps

Suppose $b, c \in \mathbb {R}$. Define $T: \mathbb {R}^{3} \rightarrow \mathbb {R}^{2}$ by

Show that $T$ is linear if and only if $b = c = 0$.

Suppose $b, c \in \mathbb {R}$. Define $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathbb {R}^{2}$ by

Show that $T$ is linear if and only if $b = c = 0$.

Suppose that $T \in \mathcal{L}\left(\mathbf{F}^{n}, \mathbf{F}^{m}\right)$. Show that there exist scalars $A_{j, k} \in \mathbf{F}$ for $j = 1, \ldots , m$ and $k = 1, \ldots , n$ such that

for every $\left(x_{1}, \ldots , x_{n}\right) \in \mathbf{F}^{n}$. This exercise shows that the linear map $T$ has the form promised in the second to last item of Example 3.3.

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

Prove that $\mathcal{L}(V, W)$ is a vector space, as was asserted in 3.6.

Prove that multiplication of linear maps has the associative, identity, and distributive properties asserted in 3.8.

Theorem 3.8 (algebraic properties of products of linear maps): associativity: $\left(T_{1} T_{2}\right) T_{3} = T_{1}\left(T_{2} T_{3}\right)$ whenever $T_{1}, T_{2}$, and $T_{3}$ are linear maps such that the products make sense (meaning $T_{3}$ maps into the domain of $T_{2}$, and $T_{2}$ maps into the domain of $T_{1}$). Identity: $T I = I T = T$ whenever $T \in \mathcal{L}(V, W)$; here the first $I$ is the identity operator on $V$, and the second $I$ is the identity operator on $W$. Distributive properties: $\left(S_{1}+S_{2}\right) T = S_{1} T+S_{2} T$ and $S\left(T_{1}+T_{2}\right) = S T_{1}+S T_{2}$ whenever $T, T_{1}, T_{2} \in \mathcal{L}(U, V)$ and $S, S_{1}, S_{2} \in \mathcal{L}(V, W)$.

Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname {dim} V = 1$ and $T \in \mathcal{L}(V)$, then there exists $\lambda \in \mathbf{F}$ such that $T v = \lambda v$ for all $v \in V$.

Give an example of a function $\varphi : \mathbb {R}^{2} \rightarrow \mathbb {R}$ such that

for all $a \in \mathbb {R}$ and all $v \in \mathbb {R}^{2}$ but $\varphi$ is not linear. This exercise and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.

Give an example of a function $\varphi : \mathbb {C} \rightarrow \mathbb {C}$ such that

for all $w, z \in \mathbb {C}$ but $\varphi$ is not linear. (Here $\mathbb {C}$ is thought of as a complex vector space.)

There also exists a function $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ such that $\varphi$ satisfies the additivity condition above but $\varphi$ is not linear. However, showing the existence of such a function involves considerably more advanced tools.

Prove or give a counterexample: If $q \in \mathcal{P}(\mathbb {R})$ and $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathcal{P}(\mathbb {R})$ is defined by $T p = q \circ p$, then $T$ is a linear map.

Suppose $V$ 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 $S \in \mathcal{L}(V)$.

Suppose $U$ is a subspace of $V$ with $U \neq V$. Suppose $S \in \mathcal{L}(U, W)$ and $S \neq 0$ (which means that $S u \neq 0$ for some $u \in U$). Define $T: V \rightarrow W$ by

Prove that $T$ is not a linear map on $V$.

Suppose $V$ is finite-dimensional. Prove that every linear map on a subspace of $V$ can be extended to a linear map on $V$. In other words, show that if $U$ is a subspace of $V$ and $S \in \mathcal{L}(U, W)$, then there exists $T \in \mathcal{L}(V, W)$ such that $T u = S u$ for all $u \in U$.

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 0$, and suppose $W$ is infinitedimensional. Prove that $\mathcal{L}(V, W)$ is infinite-dimensional.

Suppose $v_{1}, \ldots , v_{m}$ is a linearly dependent list of vectors in $V$. Suppose also that $W \neq \left\{ 0\right\}$. Prove that there exist $w_{1}, \ldots , w_{m} \in W$ such that no $T \in \mathcal{L}(V, W)$ satisfies $T v_{k} = w_{k}$ for each $k = 1, \ldots , m$.

Suppose $V$ is finite-dimensional with $\operatorname {dim} V > 1$. Prove that there exist $S, T \in \mathcal{L}(V)$ such that $S T \neq T S$.

Suppose $V$ is finite-dimensional. Show that the only two-sided ideals of $\mathcal{L}(V)$ are $\left\{ 0\right\}$ and $\mathcal{L}(V)$.