# Thread: dimensions U, W, U+W

1. ## dimensions U, W, U+W

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

2. Originally Posted by jayshizwiz
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
U=span{(1,1,1)}
W=span{(2,2,2), (1,0,0),(0,1,0)}

3. Thanks Also sprach Zarathustra for answering all my questions in all the different math sections.

4. $\displaystyle U=span(\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots ,\vec{s}_t) ~,~ W=span(\vec{w}_1,\cdots,\vec{w}_q,\vec{s}_1,\cdots ,\vec{s}_t) ~,~ U\cap W = span(\vec{s}_1,\cdots,\vec{s}_t)$

$\displaystyle dim(U) = p+t, dim(W) = q+t, dim(U+W) = p+t+q, dim(U\cap W) = t,$

$\displaystyle dim(U+W) + dim(U\cap W) = dim(U)+ dim(W)$

5. Originally Posted by math2009
$\displaystyle U=span(\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots ,\vec{s}_t) ~,~ W=span(\vec{w}_1,\cdots,\vec{w}_q,\vec{s}_1,\cdots ,\vec{s}_t) ~,~ U\cap W = span(\vec{s}_1,\cdots,\vec{s}_t)$

$\displaystyle dim(U) = p+t, dim(W) = q+t, dim(U+W) = p+t+q, dim(U\cap W) = t,$

$\displaystyle dim(U+W) + dim(U\cap W) = dim(U)+ dim(W)$
how do we know for sure dim(U) = p+t? There may be a chance that in $\displaystyle (\vec{v}_1,\cdots,\vec{v}_p,\vec{s}_1,\cdots,\vec{ s}_t)$ one of the vectors is a linear combination of the others. But I guess I understand the general thing you were saying... Thanks

6. $\displaystyle \vec{v}_i, \vec{w}_i,\vec{s}_i$ are linear independent.
They form basis of U,W