linear algebra

Problem Suppose $$U = \lbrace (x,x,y,y) \in \mathbb{F}^4 |x,y \in \mathbb{F} \rbrace$$ Find a subspace $W$ of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U \oplus W$ Solution Let $W = \lbrace (0,x^\prime,0,y^\prime) \in \mathbb{F}^4 | x^\prime,y^\prime \in \mathbb{F} \rbrace$, then: $$U + W \stackrel{D1.36}{=} \lbrace (x,x,y,y) + (0,x^\prime,0,y^\prime) | x,y,x^\prime,y^\prime \in \mathbb{F} \rbrace$$ $$\stackrel{D1.12}{=} \lbrace (x,x + x^\prime,y,y + y^\prime) | x,y,x^\prime,y^\prime \in \mathbb{F} \rbrace$$ Define $x_1 = x, x_2 = x + x^\prime, x_3 = y$ and $x_4 = x + y^\prime$, then: $$U + W = \lbrace (x_1, x_2, x_3, x_4) | x_1, x_2, x_3, x_4 \in \mathbb{F} \rbrace = \mathbb{F}^4$$ Also it’s clear that $U \cap W = \lbrace 0 \rbrace$, thus $U \oplus W = \mathbb{F}^4$....

Problem Prove or give counterexample: if $U_1$, $U_2$, $W$ are subspaces of $V$ such that $$U_1 + W = U_2 + W$$ then $U_1 = U_2$ Solution Here’s a counterexample: let $W = \lbrace (x,y,0) | x,y \in \mathbb{F} \rbrace$, $U_1 = \lbrace (0,0,z) | z \in \mathbb{F} \rbrace$ and $U_2 = \lbrace (0,z,z) | z \in \mathbb{F} \rbrace$, then by example 1.43: $$U_1 + W \stackrel{D1.36}{=} \mathbb{F}^3$$ But we also have: $$U_2 + W \stackrel{D1....

Problem Does the operation of addition on the subspaces of $V$ have an additive identity? Which subspaces have additive inverses? Solution Yes, addition of subspaces does have an additive identity since $\lbrace 0 \rbrace$ is always a subspace of any vector space $V$ and has the property that if $U$ is a subspace of $V$, then $U + \lbrace 0 \rbrace \stackrel{D1.36}{=} \lbrace u + v | u \in U, v \in \lbrace 0 \rbrace \rbrace = \lbrace u | u \in U \rbrace = U$....

Problem Is the operation of addition on the subspaces of $V$ associative? In other words, if $U_1, U_2, U_3$ are subspaces of $V$, is $$(U_1 + U_2) + U_3 = U_1 + (U_2 + U_3) ?$$ Solution Yes, Proof: $$(U_1 + U_2) + U_3 \stackrel{D1.36}{=} \lbrace u_1 + u_2 | u_1 \in U_1, u_2 \in U_2 \rbrace + U_3$$ $$\stackrel{D1.36 \land D1.39}{=} \lbrace (u_1 + u_2) + u_3 | u_1 \in U_1, u_2 \in U_2, u_3 \in U_3 \rbrace$$ $$\stackrel{D1....

Problem Is the operation of addition on the subspaces of $V$ commutative? In other words, if $U$ and $W$ are subspaces of $V$, is $U + W = W + U$ ? Solution Yes, proof: $$U + W \stackrel{D1.36}{=} \lbrace u + w | u \in U, w \in W \rbrace = \stackrel{D1.19}{=} \lbrace w + u | w \in W, u \in U \rbrace \stackrel{D1.36}{=} W + U$$