Suppose that A is an 8x11 matrix whose kernel is of dimension 5, and B is an 11x9 matrix whose image is of dimension 7. If the subspace kernel(A) + image(B) has dimension 10, what is the rank of AB?
Rank Nullity Theorem: For an n x m matrix A, dim(ker(A)) + dim(img(A)) = m
The Attempt at a Solution
Ok, so we know the following about A and B.
dim[ker(A)] = 5 (given)
dim[img(A)] = 6 (by rank-nullity theorem)
dim[img(B)] = 7 (given)
dim[ker(B)] = 2 (by rank-nullity theorem)
AB will be an 8x9 matrix, so the dimension of the image (which is the rank) can't be more than 8. I suppose the kernel of A is contained in AB, but I don't know what to make of that or how to use that information.
For the subspace ker(A) + img(B), could it be that ker(A) and img(B) share two components because the dimension is 10, but dim[ker(A)] + dim[img(B)] = 12?
This seems like a simple enough problem, but I have problems utilizing the meaning of kernel and image.