Solution 1c 20

Suppose

U = \{(x,x,y,y) \in \mathbf{F}^4 : x,y \in \mathbf{F}\}

Find a subspace $W$ of $\mathbf F^4$ such that $\mathbf F^4 = U \oplus W$ .

Let

W = \{(z,0,t,0) \in \mathbf{F}^4 : z,t \in \mathbf{F}\}

To see it's a direct sum we use 1.44, suppose $u + w = 0$ then

(x+z,x,y+t,y) = 0

Since each coordinate must be zero this implies $x=y=0$ , plugging that back in we see $z=t=0$ as well. Thus $U \cap W = \{0\}$ so by 1.44 it's a direct sum.

To see $U + W = \mathbf F^4$ is straightforward, just write an element of $\mathbf F^4$ in terms of its coordinates then solve for $x,y,z,t$ .