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

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- Jul 7th 2010, 10:20 PMjayshizwizdimensions 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 - Jul 7th 2010, 10:33 PMAlso sprach Zarathustra
- Jul 7th 2010, 10:39 PMjayshizwiz
Thanks Also sprach Zarathustra for answering all my questions in all the different math sections.

- Jul 8th 2010, 01:29 AMmath2009
$\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) $ - Jul 8th 2010, 01:43 AMjayshizwiz
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

- Jul 8th 2010, 02:02 AMmath2009
$\displaystyle \vec{v}_i, \vec{w}_i,\vec{s}_i $ are linear independent.

They form basis of U,W