Any help is appreciated! (Rofl)

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- Apr 13th 2010, 07:21 PMDarKLargest and smallest possible dimension?
Any help is appreciated! (Rofl)

- Apr 14th 2010, 02:53 AMHallsofIvy
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 PMDarK
- Apr 15th 2010, 06:08 AMHallsofIvy
A "5 by 7 matrix" has 5 rows and 7 columns. That means it is a linear operator from $\displaystyle R^7$ to $\displaystyle R^5$.

The null space is a subspace of the "domain", $\displaystyle R^7$ 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 $\displaystyle R^5$, 5. In that case the nullity would be 7- 5= 2.