I have a question here on direct sums which states:
Let U and V be subspaces of R5 such that U + V = R5, dimU = 2, and dimV = 3. Show that R5 = U is a direct sum of V.
This is how I approached it:
R5 = U is a direct sum of V
dim(U+V)= 5 = dimR5
If U is contained in V, dimU = dimV => U=V
Therefore: U+V is in R5
R5 = U is a direct sum of V.
Does that make any sense? Could someone please review this and tell me if I am correct, or atleast steer me on the right path.