dimensions U, W, U+W

**jayshizwiz** · July 7th 2010, 10:20 PM

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

**Also sprach Zarathustra** · July 7th 2010, 10:33 PM

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)}

**jayshizwiz** · July 7th 2010, 10:39 PM

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

**math2009** · July 8th 2010, 01:29 AM

$ 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) $

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

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

**jayshizwiz** · July 8th 2010, 01:43 AM

Originally Posted by math2009

$ 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) $

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

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

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

**math2009** · July 8th 2010, 02:02 AM

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

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