Problem Suppose that $U$ and $W$ are subspaces of $\mathbb{R}^{8}$ such that $\dim U = 3$, $\dim W = 5$, and $U + W = \mathbb{R}^{8}$. Prove that $\mathbb{R}^8 = U \oplus W$ Solution By theorem 2.43 we have: $$8 = \dim \mathbb{R}^{8} = \dim(U + W) $$ $$= \dim U + \dim W - \dim (U_1 \cap U_2)$$ $$= 3 + 5 - \dim(U \cap W)$$ Implying that $\dim(U \cap W) = 0$, which in turn implies $U \cap W = \lbrace 0 \rbrace$....
Problem Suppose $p_0, p_1, \dots, p_m \in \mathcal{P}(\mathbb{F})$ are such that each $p_j$ has degree $j$. Prove that $p_0, p_1, \dots, p_m$ is a basis of $\mathcal{P}_m(\mathbb{F})$ Solution Proceed by induction on $m$. BC: Let $m = 0$, then $p_0 = a_0z^0 = a_0$ is a basis for $\textrm{span}(a_0) = \textrm{span}(1) = \mathcal{P}_0(\mathbb{F})$. IH: Suppose that for some $m = k \in \mathbb{N}$ we have $p_0, p_1, \dots, p_k$ is a basis for $\mathcal{P}_k(\mathbb{F})$ if $p_0, p_1, \dots, p_k \in \mathcal{P}(\mathbb{F})$ are such that each $p_j$ has degree $j$....
Problem Suppose $v_1, \dots, v_m$ is linearly independent in $V$ and $w \in V$. Prove that $$\dim \textrm{span}(v_1 + w, \dots, v_m + w) \geq m - 1$$ Solution There are two cases, either $w \in \textrm{span}(v_1, \dots, v_m)$ or it is not. Given that $w$ $\cancel{\in}$ $\textrm{span}(v_1, \dots, v_m)$, then P2A10 implies $\dim \textrm{span}(v_1 + w, \dots, v_m + w) = m$ - this is the contrapositive of the problem statement in the exercise....
Problem (a) Let $U = \lbrace p \in \mathcal{P}_4(\mathbb{F}) | \int_x^y p = 0 \land -x=y=1\rbrace$. Find a basis of $U$ (b) Extend the basis in part (a) to a basis of $\mathcal{P}_4(\mathbb{F})$ (c) Find a subspace $W$ of $\mathcal{P}_4(\mathbb{F})$ such that $\mathcal{P}_4(\mathbb{F}) = U \oplus W$ Solution (a) It’s clear that none of the vectors in the linearly independent list $1, z^2, z^4$ are in $U$, therefore by theorem 2....
Problem (a) Let $U = \lbrace p \in \mathcal{P}_4(\mathbb{F}) | p(2) = p(5) = p(6) \rbrace$. Find a basis of $U$ (b) Extend the basis in part (a) to a basis of $\mathcal{P}_4(\mathbb{F})$ (c) Find a subspace $W$ of $\mathcal{P}_4(\mathbb{F})$ such that $\mathcal{P}_4(\mathbb{F}) = U \oplus W$ Solution (a) $U$ is a subspace of $U^\prime = \lbrace p \in \mathcal{P}_4(\mathbb{F}) | p(2) = p(5) \rbrace$, as any vector in $U$ is also in $U^\prime$....