Exercises 2B $ \begin{array}{l} \text{Exercises 2B} \\ \\ 1 \text{ Find all vector spaces that have exactly one basis.} \\ \\ 2 \text{ Verify all assertions in Example 2.27.} \\ \\ 3 \text{(a) Let } U \text{ be the subspace of } \mathbb{R}^5 \text{ defined by} \\ U = \{(x_1, x_2, x_3, x_4, x_5) \in \mathbb{R}^5 : x_1 = 3x_2 \text{ and } x_3 = 7x_4\}. \\ \text{Find a basis of } U. \\\\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{R}^5. \\\\ (c) \text{Find a subspace } W \text{ of } \mathbb{R}^5 \text{ such that } \mathbb{R}^5 = U \oplus W. \\\\ 4 \text{(a) Let } U \text{ be the subspace of } \mathbb{C}^5 \text{ defined by } U = \{(z_1, z_2, z_3, z_4, z_5) \in \mathbb{C}^5 : 6z_1 = z_2 \text{ and } z_3 + 2z_4 + 3z_5 = 0\}. \\ \text{Find a basis of } U. \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{C}^5. \\ (c) \text{Find a subspace } W \text{ of } \mathbb{C}^5 \text{ such that } \mathbb{C}^5 = U \oplus W. \\ \\\\ 5 \text{ Suppose } V \text{ is finite-dimensional and } U, W \text{ are subspaces of } V \text{ such that } V = U + W. \\ \text{Prove that there exists a basis of } V \text{ consisting of vectors in } U \cup W. \\ \\ 6 \text{ Prove or give a counterexample: If } p_0, p_1, p_2, p_3 \text{ is a list in } \mathcal{P}_3(\mathbb{F}) \text{ such that none of the polynomials} \\ p_0, p_1, p_2, p_3 \text{ has degree 2, then } p_0, p_1, p_2, p_3 \text{ is not a basis of } \mathcal{P}_3(\mathbb{F}). \\ \\ 7 \text{ Suppose } v_1, v_2, v_3, v_4 \text{ is a basis of } V. \text{ Prove that} \\ v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4 \text{ is also a basis of } V. \\ \\ 8 \text{ Prove or give a counterexample: If } v_1, v_2, v_3, v_4 \text{ is a basis of } V \text{ and } U \text{ is a subspace of } V \text{ such that} \\ v_1, v_2 \in U \text{ and } v_3 \notin U \text{ and } v_4 \notin U, \text{ then } v_1, v_2 \text{ is a basis of } U. \\ \\ 9 \text{ Suppose } v_1, \ldots, v_m \text{ is a list of vectors in } V. \text{ For } k \in \{1, \ldots, m\}, \text{ let } w_k = v_1 + \ldots + v_k. \\ \text{ Show that } v_1, \ldots, v_m \text{ is a basis of } V \text{ if and only if } w_1, \ldots, w_m \text{ is a basis of } V. \\ \\ 10 \text{ Suppose } U \text{ and } W \text{ are subspaces of } V \text{ such that } V = U \oplus W.\\ \text{ Suppose also that } u_1, \ldots, u_m \text{ is a basis of } U \\ \text{ and } w_1, \ldots, w_n \text{ is a basis of } W. \text{ Prove that } u_1, \ldots, u_m, w_1, \ldots, w_n \text{ is a basis of } V. \\ \\ 11 \text{ Suppose } V \text{ is a real vector space. Show that if } v_1, \ldots, v_n \text{ is a basis of } V \text{ (as a real vector space), then} \\ v_1, \ldots, v_n \text{ is also a basis of the complexification } V_{\mathbb{C}} \text{ (as a complex vector space).} \\ \text{See Exercise 8 in Section 1B for the definition of the complexification } V_{\mathbb{C}}. \end{array} $