Chapter 4.11, Problem 36AP

Let $\left(V,+v,\cdot v\right)$ and $\left(W,+w,\cdot w\right)$ be vector spaces and define

$V\oplus W=\left\{\left(v,w\right):v\in V\text{and}\text{w}\in W\right\}$.

Prove that

(a) $V\oplus W$ is a vector space, under componentwise operations.

(b) if $S=\left\{\left(v,0\right):v\in V\right\}$ and $S^{\prime }=\left\{\left(0,w\right):w\in W\right\}$, then $S$ and $S^{\prime }$ are subspaces of $V\oplus W$.

(c) if $\dim \left[V\right]=n$ and $\dim \left[W\right]=m$, then $\dim \left[V\oplus W\right]=m+n$.

[Hint: Write a basis for $V\oplus W$ in terms of bases for $V$ and $W$.]