Question

If U and W are finite dimensional subspaces of a vector Space V then dim(U+W)=​

Answer 1

We need to recall the following Grassmann's Identity.

• $A$ and $B$ are finite-dimensional subspaces of a vector space $V$, then $dim(A+B)=dim(A)+dim(B)-dim(A\cap B)$

Given:

$U$ and $W$ are finite-dimensional subspaces of a vector space $V$.

Using Grassmann's Identity, we get

The dimension of subspaces $U+W$ is,
$dim(U+W)=dim(U)+dim(W)-dim(U\cap W)$