Hi there just something I'm a bit unsure of and want it cleared up if you can help me please?
So in my book it gives a 3x5 matrix, and it tells me that rank(A) is always less than or equal to 3, can this be generalised to a mxn matrix, with rank less than or equal to n, where m>n, and rank less than or equal to m, if m<n?
Also, I don't understand how the rank can be zero? Would this not mean that there are no vectors in the basis; is this possible?
yes, a matrix can have rank 0. this one does:
[0 0]
[0 0]. no choice of columns of this matrix leads to a basis (0-vectors are ALWAYS linearly dependent, since even if a is non-zero a0 = 0), so the column space has the "empty basis", {Ø}, with 0 elements.