let's say dimU = 1 and dimW = 3
does dim(U+W) always equal dimU + dim W?
If not, can someone please show me a counter example?
Thankssssssss
$\displaystyle
U=span(\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots ,\vec{s}_t) ~,~
W=span(\vec{w}_1,\cdots,\vec{w}_q,\vec{s}_1,\cdots ,\vec{s}_t) ~,~
U\cap W = span(\vec{s}_1,\cdots,\vec{s}_t)
$
$\displaystyle dim(U) = p+t, dim(W) = q+t, dim(U+W) = p+t+q, dim(U\cap W) = t, $
$\displaystyle dim(U+W) + dim(U\cap W) = dim(U)+ dim(W) $