Direct Sum of Two Subspaces If $U = \{x, y, x+y, x-y, 2x\}$ is a subspace, find a subspace $W$ of $F^5$ such that $F^5 = U+W$ is a direct sum.
So if $U+W$ is a direct sum then their intersection must be $\{0\}$, and that the representation of any sum of their elements is unique. I tried approaching this problem using the intersection being $\{0\}$ but I just don't know how to go about the math, can anyone help me?
 A: $U=\begin{pmatrix} x \\ y\\ x+y\\ x-y\\ 2x \end{pmatrix}=\begin{pmatrix} x \\ 0 \\ x \\ x \\ 2x\end{pmatrix}+\begin{pmatrix} 0 \\ y \\ y \\ -y \\ 0 \end{pmatrix}=span\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \\2 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\0 \end{pmatrix}\}$
It is sufficient to find a single vector $v \in F^5$ such that $v \notin U$, and then define $W=span\{v\}$
Can you find such a vector $v$?
A: Note that $U$ as you defined is just span of $(1,0,0,0,0)$ and $(0,1,0,0,0)$ thus you can let $W$ just be span of $(0,0,1,0,0), (0,0,0,1,0)$ and $(0,0,0,0,1)$
To help you get to these answers in generally it is a good idea to find the dimension and thus basis from this and then W is span of basis vectors of the whole vector space that are not included in basis of U
Note I assumed you meant that $U=\{x,y,x+y,x-y,2x\}$ as $span\{x,y,x+y,x-y,2x\}$ 
A: For those who wonder how to get $3$ linearly independent vectors which are not in $U$, here comes a brute method:

1) Find the determinant of the matrix $A$:

$$
    A=\begin{pmatrix}
    1 & 0 & 1 & 1 & 2 \\
    0 & 1 & 1 & -1 & 0 \\
    a & b & c & d & e \\
    f  & g & h & i & j \\
    k  & l  & m & n & o \\
\end{pmatrix}
$$
in function of the $15$ constants (which are real numbers). It is note as difficult as it may seem to the untrained eye; use Laplace expansion.   

2) Determine for which values of the constants the determinant is non-zero. 

We do this because we are interested in finding a unique solution (which implies setting the five rows to be linearly independent) for the system described by the matrix $A$.
You will end up with $3$ vectors that span $W$, as required. 
NOTE: Of course the 2 vectors spanning $W$ are not unique. 
