$ \begin{array}{l} \text{Exercises 3A} \\ \\ 1 \text{ Suppose } b, c \in \mathbb{R}. \text{ Define } T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \text{ by} \\ T(x, y, z) = (2x - 4y + 3z + b, 6x + cxyz). \\ \text{Show that } T \text{ is linear if and only if } b = c = 0. \\ \\ 2 \text{ Suppose } b, c \in \mathbb{R}. \text{ Define } T : \mathbb{P}(\mathbb{R}) \rightarrow \mathbb{R}^2 \text{ by} \\ T(p) = (3p(4) + 5p'(6) + bp(1)p(2), \int_{-1}^{2} x^3p(x) dx + c \sin p(0)). \\ \text{Show that } T \text{ is linear if and only if } b = c = 0. \\ \\ 3 \text{ Suppose that } T \in L(\mathbb{F}^n,\mathbb{F}^m). \text{ Show that there exist scalars } A_{j,k} \in \mathbb{F} \text{ for} \\ j = 1,...,m \text{ and } k = 1,...,n \text{ such that} \\ T(x_1,...,x_n) = (A_{1,1}x_1 + \cdots + A_{1,n}x_n,...,A_{m,1}x_1 + \cdots + A_{m,n}x_n) \\ \text{for every } (x_1, ..., x_n) \in \mathbb{F}^n. \\ \text{This exercise shows that the linear map } T \text{ has the form promised in the} \\ \text{second to last item of Example 3.3.} \\ \\ 4 \text{ Suppose } T \in L(V,W) \text{ and } v_1,...,v_m \text{ is a list of vectors in } V \text{ such that } T v_1, ..., T v_m \text{ is a linearly independent list in } W. \text{ Prove that } v_1, ..., v_m \text{ is linearly independent.} \\ \\ 5 \text{ Prove that } L(V, W) \text{ is a vector space, as was asserted in 3.6.} \\ \\ 6 \text{ Prove that multiplication of linear maps has the associative, identity, and} \\ \text{distributive properties asserted in 3.8.} \\ \\ 7 \text{ Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if } \text{dim } V = 1 \text{ and } T \in L(V), \text{ then there exists } \lambda \in \mathbb{F} \text{ such that } T v = \lambda v \text{ for all } v \in V. \\ \\ 8 \text{ Give an example of a function } \phi : \mathbb{R}^2 \rightarrow \mathbb{R} \text{ such that } \phi(av) = a\phi(v) \\ \text{for all } a \in \mathbb{R} \text{ and all } v \in \mathbb{R}^2 \text{ but } \phi \text{ is not linear.} \\ \text{This exercise and the next exercise show that neither homogeneity nor} \\ \text{additivity alone is enough to imply that a function is a linear map.} \\ \\ 9 \text{ Give an example of a function } \phi : \mathbb{C} \rightarrow \mathbb{C} \text{ such that } \phi(w+z) = \phi(w) + \phi(z) \\ \text{for all } w, z \in \mathbb{C} \text{ but } \phi \text{ is not linear. (Here } \mathbb{C} \text{ is thought of as a complex vector space.)} \\ \text{There also exists a function } \phi : \mathbb{R} \rightarrow \mathbb{R} \text{ such that } \phi \text{ satisfies the additivity condition above but } \phi \text{ is not linear. However, showing the existence of such a function involves considerably more advanced tools.} \\ \\ 10 \text{ Prove or give a counterexample: If } q \in \mathbb{P}(\mathbb{R}) \text{ and } T : \mathbb{P}(\mathbb{R}) \rightarrow \mathbb{P}(\mathbb{R}) \text{ is} \\ \text{defined by } Tp = q \circ p, \text{ then } T \text{ is a linear map.} \\ \text{The function } T \text{ defined here differs from the function } T \text{ defined in the last} \\ \text{bullet point of 3.3 by the order of the functions in the compositions.} \\ \\ 11 \text{ Suppose } V \text{ is finite-dimensional and } T \in L(V). \text{ Prove that } T \text{ is a scalar} \\ \text{multiple of the identity if and only if } ST = TS \text{ for every } S \in L(V). \\ \\ 12 \text{ Suppose } U \text{ is a subspace of } V \text{ with } U \neq V. \text{ Suppose } S \in L(U,W) \text{ and} \\ S \neq 0 \text{ (which means that } Su \neq 0 \text{ for some } u \in U). \text{ Define } T : V \rightarrow W \text{ by } \\ \begin{cases} Sv & \text{if } v \in U, \\ 0 & \text{if } v \in V \text{ and } v \notin U. \end{cases} \\ \text{ Prove that } T \text{ is not a linear map on } V. \\ \\ 13 \text{ Suppose } V \text{ is finite-dimensional. Prove that every linear map on a subspace of } V \text{ can be extended to a linear map on } V. \text{ In other words, show that if } U \text{ is a subspace of } V \text{ and } S \in L(U, W), \text{ then there exists } T \in L(V, W) \text{ such that } Tu = Su \text{ for all } u \in U. \\ \text{The result in this exercise is used in the proof of 3.125.} \\ \\ 14 \text{ Suppose } V \text{ is finite-dimensional with dim } V > 0, \text{ and suppose } W \text{ is infinite-dimensional. Prove that } L(V, W) \text{ is infinite-dimensional.} \\ \\ 15 \text{ Suppose } v_1, ..., v_m \text{ is a linearly dependent list of vectors in } V. \text{ Suppose also that } W \neq \{0\}. \text{ Prove that there exist } w_1, ..., w_m \in W \text{ such that no } T \in L(V, W) \text{ satisfies } Tv_k = w_k \text{ for each } k = 1, ..., m. \\ \\ 16 \text{ Suppose } V \text{ is finite-dimensional with dim } V > 1. \text{ Prove that there exist } S,T \in L(V) \text{ such that } ST \neq TS. \\ \\ 17 \text{ Suppose } V \text{ is finite-dimensional. Show that the only two-sided ideals of } L(V) \text{ are } \{0\} \text{ and } L(V). \\ \text{A subspace } E \text{ of } L(V) \text{ is called a two-sided ideal of } L(V) \text{ if } Tv \in E \text{ and } ET \in E \text{ for all } E \in E \text{ and all } T \in L(V). \end{array} $