direct sums with different dimensions

Question

Is it possible for two subspaces of a different dimension to have a direct sum?

For example the vectorspace ${R}$ [X]<3 with the subspaces U=${R}$ [X]<1 and W={a+aX+bX^2+bX^3|a,b $\in$ ${R}$}.

Answer 1

Consider, as you said $V = \def\R{\mathbf R}\R[X]_{\le 3}$, with the subspaces

$U = \R[X]_{\le 1}$, and $W = \{a+aX + bX^2 + bX^3: a,b\in \mathbf R\}$.

This sum is not direct, as $1 + X \in U \cap W$.

But there are direct sums of spaces with different dimensions:

$U = \R[X]_{\le 2}$ and $W = \{aX^3: a \in \R\}$.

Then $V = U \oplus W$: For $p = a+bX+cX^2+dX^3\in V$, we have $$ p = (a+bX+cX^2) + dX^3 \in U + W $$ On the other hand if $p=a+bX+cX^2+dX^3 \in U \cap W$, we have $d = 0$ due to $p \in U$ and $a=b=c = 0$ due to $p \in W$. Hence $p = 0$.

So, this sum is direct, but $\dim U = 3$, $\dim W = 1$. As allways for direct sums, we have $$ \dim(U \oplus W) = \dim U + \dim W = 3+ 1 = 4. $$