Exercise 28

(Requires the subsection on Combining Subspaces, which is optional.) Let $U$ and $W$ be vector spaces. Define a new vector space, consisting of the set $U\times W=\text{$\{\hskip 1.00006pt(\vec{u},\vec{w})\mid\vec{u}\in U\text{\ and\ }\vec{w}\in W\hskip 1.00006pt\}$}$ along with these operations.

This is a vector space, the external direct sum of $U$ and $W$.

• Check that it is a vector space.

• Find a basis for, and the dimension of, the external direct sum $\mathcal{P}_{2}\times\mathbb{R}^{2}$.

• What is the relationship among $\dim(U)$, $\dim(W)$, and $\dim(U\times W)$?

• Suppose that $U$ and $W$ are subspaces of a vector space $V$ such that $V=U\oplus W$ (in this case we say that $V$ is the internal direct sum of $U$ and $W$). Show that the map $f\colon U\times W\to V$ given by

  $$(\vec{u},\vec{w})\stackrel{{\scriptstyle f}}{{\longmapsto}}\vec{u}+\vec{w}$$

  is an isomorphism. Thus if the internal direct sum is defined then the internal and external direct sums are isomorphic.