$ \begin{array}{l} \text{ Exercises 2C} \\ \\ 1 \text{ Show that the subspaces of } \mathbb{R}^2 \text{ are precisely } \{0\}, \text{ all lines in } \mathbb{R}^2 \text{ containing the origin, and } \mathbb{R}^2. \\ \\ 2 \text{ Show that the subspaces of } \mathbb{R}^3 \text{ are precisely } \{0\}, \text{ all lines in } \mathbb{R}^3 \text{ containing the origin, all planes in } \mathbb{R}^3 \text{ containing the origin, and } \mathbb{R}^3. \\ \\ 3 \text{(a) Let } U = \{p \in \mathbb{P}_4(\mathbb{F}) : p(6) = 0\}. \text{ Find a basis of } U. \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{P}_4(\mathbb{F}). \\ (c) \text{Find a subspace } W \text{ of } \mathbb{P}_4(\mathbb{F)} \text{ such that } \mathbb{P}_4(\mathbb{F)} = U \oplus W. \\ \\ 4 \text{(a) Let } U = \{p \in \mathbb{P}_4(\mathbb{R}) : p''(6) = 0\}. \text{ Find a basis of } U. \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{P}_4 (\mathbb{R} ). \\ (c) \text{Find a subspace } W \text{ of } \mathbb{P}_4(\mathbb{R}) \text{ such that } \mathbb{P}_4(\mathbb{R}) = U \oplus W. \\ \\ 5 \text{(a) Let } U = \{p \in \mathbb{P}_4(\mathbb{F}) : p(2) = p(5)\}. \text{ Find a basis of } U. \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{P}_4(\mathbb{F}). \\ (c) \text{Find a subspace } W \text{ of } \mathbb{P}_4(\mathbb{F)} \text{ such that } \mathbb{P}_4(\mathbb{F)} = U \oplus W. \\ \\ 6 \text{(a) } \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{P}_4(\mathbb{F}). \\ (c) \text{Find a subspace } W \text{ of } \mathbb{P}_4(\mathbb{F)} \text{ such that } \mathbb{P}_4(\mathbb{F)} = U \oplus W. \\ \\ 7 \text{(a) } \\ (b) \text{Extend the basis in (a) to a basis of } \mathbb{P}_4 (\mathbb{R} ). \\ (c) \text{Find a subspace } W \text{ of } \mathbb{P}_4(\mathbb{R}) \text{ such that } \mathbb{P}_4(\mathbb{R}) = U \oplus W. \\ \\ 8 \text{ Suppose } v_1, ..., v_m \text{ is linearly independent in } V \text{ and } w \in V. \text{ Prove that } \text{dim span}(v_1 + w,...,v_m + w) \geq m - 1. \\ \\ 9 \text{ Suppose } m \text{ is a positive integer and } p_0, p_1, ..., p_m \in \mathbb{P}(\mathbb{F}) \text{ are such that each } p_k \text{ has degree } k. \text{ Prove that } p_0, p_1, ..., p_m \text{ is a basis of } \mathbb{P}_m(\mathbb{F}). \\ \\ 10 \text{ Suppose } m \text{ is a positive integer. For } 0 \leq k \leq m, \text{ let } p_k(x) = x^k(1 - x)^{m-k}. \\ \text{Show that } p_0, ..., p_m \text{ is a basis of } \mathbb{P}_m(\mathbb{F}). \\ \text{The basis in this exercise leads to what are called Bernstein polynomials. You can do a web search to learn how Bernstein polynomials are used to approximate continuous functions on } [0, 1]. \\ \\ 11 \text{ Suppose } U \text{ and } W \text{ are both four-dimensional subspaces of } \mathbb{C}^6. \text{ Prove that there exist two vectors in } U \cap W \text{ such that neither of these vectors is a scalar multiple of the other.} \\ \\ 12 \text{ Suppose that } U \text{ and } W \text{ are subspaces of } \mathbb{R}^8 \text{ such that } \text{dim } U = 3, \text{dim } W = 5, \text{ and } U + W = \mathbb{R}^8. \text{ Prove that } \mathbb{R}^8 = U \oplus W. \\ \\ 13 \text{ Suppose } U \text{ and } W \text{ are both five-dimensional subspaces of } \mathbb{R}^9. \text{ Prove that } U \cap W \neq \{0\}. \\ \\ 14 \text{ Suppose } V \text{ is a ten-dimensional vector space and } V_1, V_2, V_3 \text{ are subspaces of } V \text{ with } \text{dim } V_1 = \text{dim } V_2 = \text{dim } V_3 = 7. \text{ Prove that } V_1 \cap V_2 \cap V_3 \neq \{0\}. \\ \\ 15 \text{ Suppose } V \text{ is finite-dimensional and } V_1, V_2, V_3 \text{ are subspaces of } V \text{ with } \text{dim } V_1 + \text{dim } V_2 + \text{dim } V_3 > 2\text{dim } V. \text{ Prove that } V_1 \cap V_2 \cap V_3 \neq \{0\}. \\ \\ 16 \text{ Suppose } V \text{ is finite-dimensional and } U \text{ is a subspace of } V \text{ with } U \neq V. \text{ Let } n = \text{dim } V \text{ and } m = \text{dim } U. \text{ Prove that there exist } n - m \text{ subspaces of } V, \text{ each of dimension } n - 1, \text{ whose intersection equals } U. \\ \\ 17 \text{ Suppose that } V_1, ..., V_m \text{ are finite-dimensional subspaces of } V. \text{ Prove that } V_1 + \cdots + V_m \text{ is finite-dimensional and} \\ \text{dim}(V_1 + \cdots + V_m) \leq \text{dim} V_1 + \cdots + \text{dim} V_m. \\ \text{The inequality above is an equality if and only if } V_1 + \cdots + V_m \text{ is a direct sum, as will be shown in 3.94.} \\ \\ \text{18 Suppose } V \text{ is finite-dimensional, with dim } V = n \geq 1. \text{ Prove that there exist} \\ \text{one-dimensional subspaces } V_1, ..., V_n \text{ of } V \text{ such that } V = V_1 \oplus \cdots \oplus V_n. \\ \\ \text{19 Explain why you might guess, motivated by analogy with the formula for the number of elements in the union of three finite sets, that if } V_1, V_2, V_3 \text{ are subspaces of a finite-dimensional vector space, then} \\ \text{dim}(V_1 + V_2 + V_3) = \text{dim} V_1 + \text{dim} V_2 + \text{dim} V_3 - \text{dim}(V_1 \cap V_2) - \text{dim}(V_1 \cap V_3) - \text{dim}(V_2 \cap V_3) + \text{dim}(V_1 \cap V_2 \cap V_3). \\ \text{Then either prove the formula above or give a counterexample.} \\ \\ \text{20 Prove that if } V_1,V_2, \text{ and } V_3 \text{ are subspaces of a finite-dimensional vector space, then} \\ \text{dim}(V_1 + V_2 + V_3) = \text{dim} V_1 + \text{dim} V_2 + \text{dim} V_3 - \text{dim}(V_1 \cap V_2) + \text{dim}(V_1 \cap V_3) + \text{dim}(V_2 \cap V_3) 3 \\ - \text{dim}((V_1+V_2)\cap V_3) + \text{dim}((V_1+V_3)\cap V_2) + \text{dim}((V_2 +V_3)\cap V_1). 3 \\ \text{The formula above may seem strange because the right side does not look like an integer.} \end{array} $