Let $V=k^3$ for some field $k$. Let $W$ be the subspace spanned by $(1,0,0)$ and let $U$ be the subspace spanned by $(1,1,0)$ and $(0,1,1)$. Show that $V= W \oplus U$. Explain your argument in detail.
What I Know
I know that a field $k^n=n$-tuples of elements of $k$.
I know that a subset $W$ of a vector space $V$ over a field $k$ is a subspace if the operations of $V$ make $W$ into a vector space over $k$.
I know that if span$(S)=V$ for a set $S$ in a vector space $V$, where $S$ is linearly independent, then $S$ is a basis for $V$.
I know that the external direct sum $V \oplus W$ for vector spaces $V$ and $W$ over a field $k$ is defined as the set of all ordered pairs $(v,w)$ such that $v\in V$ and $w \in W$.
What I Don't Know
How to apply what I listed above to help me solve the problem. I am absolutely atrocious at this material and struggle so much in simply starting these problems.
If everything I listed above is even relevant to the problem at hand.
If what I listed above is insufficient to complete the problem.
Text: Abstract Linear Algebra by Curtis