Any help is appreciated!
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?
A "5 by 7 matrix" has 5 rows and 7 columns. That means it is a linear operator from to .
The null space is a subspace of the "domain", 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 , 5. In that case the nullity would be 7- 5= 2.