# Largest and smallest possible dimension?

• Apr 13th 2010, 07:21 PM
DarK
Largest and smallest possible dimension?
Any help is appreciated! (Rofl)
• Apr 14th 2010, 02:53 AM
HallsofIvy
Have you given this any thought at all? Do you know the "rank- nullity" theorem: If L:U-> V, then the rank of L (dimension of the image L(U)) plus the nullity of L (dimension of null space of L) is equal to the dimension of U.

Now, what is the dimension of U?
• Apr 14th 2010, 03:52 PM
DarK
Quote:

Originally Posted by HallsofIvy
Have you given this any thought at all? Do you know the "rank- nullity" theorem: If L:U-> V, then the rank of L (dimension of the image L(U)) plus the nullity of L (dimension of null space of L) is equal to the dimension of U.

Now, what is the dimension of U?

Ok I've given thought to it.

The smallest possible dimension of the soln space of Ax = 0 is 2
And the largest possible dimension of Ax = 0 is 0? or 1?

I'm not so sure about the second.

Thanks.
• Apr 15th 2010, 06:08 AM
HallsofIvy
A "5 by 7 matrix" has 5 rows and 7 columns. That means it is a linear operator from $R^7$ to $R^5$.
The null space is a subspace of the "domain", $R^7$ so the maximum possible nullity is 7. (And the 0 matrix has the whole space as null space. In that case, the rank would be 7- 7= 0.

At the other extreme, if the the matrix could be row reduced so that none of its 5 rows are all 0, the rank would be the entire dimension of $R^5$, 5. In that case the nullity would be 7- 5= 2.