$ \begin{array}{l} \text{Exercises 3B} \\ \\ 1 \text{ Give an example of a linear map } T \text{ with } \text{dim null } T = 3 \text{ and } \text{dim range } T = 2. \\ \\ 2 \text{ Suppose } S, T \in L(V) \text{ are such that range } S \subseteq \text{ null } T. \text{ Prove that } (ST)^2 = 0. \\ \\ 3 \text{ Suppose } v_1, ..., v_m \text{ is a list of vectors in } V. \text{ Define } T \in L(F^m, V) \text{ by} \\ T(z_1,...,z_m) = z_1v_1 + \cdots + z_mv_m. \\ \text{(a) What property of } T \text{ corresponds to } v_1, ..., v_m \text{ spanning } V? \\ \text{(b) What property of } T \text{ corresponds to the list } v_1,...,v_m \text{ being linearly independent?} \\ \\ 4 \text{ Show that } \{ T \in L(\mathbb{R}^5, \mathbb{R}^4) \,|\, \text{dim null } T > 2 \} \text{ is not a subspace of } L(\mathbb{R}^5, \mathbb{R}^4). \\ \\ 5 \text{ Give an example of } T \in L(\mathbb{R}^4) \text{ such that range } T = \text{ null } T. \\ \\ 6 \text{ Prove that there does not exist } T \in L(\mathbb{R}^5) \text{ such that range } T = \text{ null } T. \\ \\ 7 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional with } 2 \leq \text{ dim } V \leq \text{ dim } W. \text{ Show that } \{ T \in L(V, W) \,|\, T \text{ is not injective} \} \text{ is not a subspace of } L(V, W). \\ \\ 8 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional with } \text{ dim } V \geq \text{ dim } W \geq 2. \text{ Show that } \{ T \in L(V, W) \,|\, T \text{ is not surjective} \} \text{ is not a subspace of } L(V, W). \\ \\ 9 \text{ Suppose } T \in L(V, W) \text{ is injective and } v_1, ..., v_n \text{ is linearly independent in } V. \text{ Prove that } T v_1, ..., T v_n \text{ is linearly independent in } W. \\ \\ 10 \text{ Suppose } v_1, ..., v_n \text{ spans } V \text{ and } T \in L(V, W). \text{ Show that } T v_1, ..., T v_n \text{ spans range } T. \\ \\ 11 \text{ Suppose that } V \text{ is finite-dimensional and that } T \in L(V, W). \text{ Prove that there exists a subspace } U \text{ of } V \text{ such that} \\ U \cap \text{null } T = \{0\} \text{ and range } T = \{T u \,|\, u \in U\}. \\ \\ 12 \text{ Suppose } T \text{ is a linear map from } \mathbb{F}^4 \text{ to } \mathbb{F}^2 \text{ such that} \\ \text{null } T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{F}^4 \,|\, x_1 = 5x_2 \text{ and } x_3 = 7x_4 \}. \text{ Prove that } T \text{ is surjective.} \\ \\ 13 \text{ Suppose } U \text{ is a three-dimensional subspace of } \mathbb{R}^8 \text{ and that } T \text{ is a linear map from } \mathbb{R}^8 \text{ to } \mathbb{R}^5 \text{ such that null } T = U. \text{ Prove that } T \text{ is surjective.} \\ \\ 14 \text{ Prove that there does not exist a linear map from } \mathbb{F}^5 \text{ to } \mathbb{F}^2 \text{ whose null space equals} \\ \{ (x_1, x_2, x_3, x_4, x_5) \in \mathbb{F}^5 \,|\, x_1 = 3x_2 \text{ and } x_3 = x_4 = x_5 \}. \\ \\ 15 \text{ Suppose there exists a linear map on } V \text{ whose null space and range are both finite-dimensional.} \\ \text{Prove that } V \text{ is finite-dimensional.} \\ \\ \text{Section 3B Null Spaces and Ranges} \\ \\ 16 \text{ Suppose } V \text{ and } W \text{ are both finite-dimensional. Prove that there exists an} \\ \text{injective linear map from } V \text{ to } W \text{ if and only if dim } V \leq \text{ dim } W. \\ \\ 17 \text{ Suppose } V \text{ and } W \text{ are both finite-dimensional. Prove that there exists a} \\ \text{surjective linear map from } V \text{ onto } W \text{ if and only if dim } V \geq \text{ dim } W. \\ \\ 18 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional and that } U \text{ is a subspace of } V. \text{ Prove that there exists } T \in L(V,W) \text{ such that null } T = U \text{ if and only if } \text{dim } U \geq \text{dim } V - \text{dim } W. \\ \\ 19 \text{ Suppose } W \text{ is finite-dimensional and } T \in L(V, W). \text{ Prove that } T \text{ is injective if and only if there exists } S \in L(W, V) \text{ such that } ST \text{ is the identity operator on } V. \\ \\ 20 \text{ Suppose } W \text{ is finite-dimensional and } T \in L(V, W). \text{ Prove that } T \text{ is surjective if and only if there exists } S \in L(W, V) \text{ such that } TS \text{ is the identity operator on } W. \\ \\ 21 \text{ Suppose } V \text{ is finite-dimensional, } T \in L(V, W), \text{ and } U \text{ is a subspace of } W. \text{ Prove that } \{ v \in V \,|\, Tv \in U \} \text{ is a subspace of } V \text{ and} \\ \text{dim} \{ v \in V \,|\, Tv \in U \} = \text{dim null } T + \text{dim} (U \cap \text{range } T). \\ \\ 22 \text{ Suppose } U \text{ and } V \text{ are finite-dimensional vector spaces and } S \in L(V, W) \text{ and} \\ T \in L(U, V). \text{ Prove that} \\ \text{dim null } ST \leq \text{dim null } S + \text{dim null } T. \\ \\ 23 \text{ Suppose } U \text{ and } V \text{ are finite-dimensional vector spaces and } S \in L(V, W) \text{ and } T \in L(U, V). \text{ Prove that} \\ \text{dim range } ST \leq \text{min}\{\text{dim range } S, \text{dim range } T\}. \\ \\ 24 \text{(a) Suppose } \text{dim } V = 5 \text{ and } S, T \in L(V) \text{ are such that } ST = 0. \text{ Prove that } \text{dim range } TS \leq 2. \\ (b) \text{ Give an example of } S, T \in L(F^5) \text{ with } ST = 0 \text{ and } \text{dim range } TS = 2. \\ \\ 25 \text{ Suppose that } W \text{ is finite-dimensional and } S, T \in L(V, W). \text{ Prove that} \\ \text{null } S \subseteq \text{null } T \text{ if and only if there exists } E \in L(W) \text{ such that } T = ES. \\ \\ 26 \text{ Suppose that } V \text{ is finite-dimensional and } S, T \in L(V, W). \text{ Prove that} \\ \text{range } S \subseteq \text{range } T \text{ if and only if there exists } E \in L(V) \text{ such that } S = TE. \\ \\ 27 \text{ Suppose } P \in L(V) \text{ and } P^2 = P. \text{ Prove that } V = \text{null } P \oplus \text{range } P. \\ \\ 28 \text{ Suppose } D \in L(\mathbb{P}(\mathbb{R})) \text{ is such that deg } Dp = (\text{deg } p) - 1 \text{ for every non-constant polynomial } p \in \mathbb{P}(\mathbb{R}). \text{ Prove that } D \text{ is surjective.} \\ \text{The notation } D \text{ is used above to remind you of the differentiation map that sends a polynomial } p \text{ to } p'. \\ \\ 29 \text{ Suppose } p \in \mathbb{P}(\mathbb{R}). \text{ Prove that there exists a polynomial } q \in \mathbb{P}(\mathbb{R}) \text{ such} \\ \text{that } 5q'' + 3q' = p. \\ \text{This exercise can be done without linear algebra, but it’s more fun to do it using linear algebra.} \\ \\ 30 \text{ Suppose } \phi \in L(V,F) \text{ and } \phi \neq 0. \text{ Suppose } u \in V \text{ is not in null } \phi. \text{ Prove that} \\ V = \text{null } \phi \oplus \{ a u \,|\, a \in F \}. \end{array} $