CS => Column Space
I first had to show that CS(AB) is a subspace of CS(A), this is true because when we multiply A*B every column vector of this new matrix AB is a linear combination of vector A.
But now I have to show that this inclusion CS(BA) sub-space of CS(A) is not always true?
When I do this example A = [0 0 , 1 2] and B = [5 7 , 0 0] then BA = [0 0, 5 7], then this means is unsolvable because there is no x[0 , 0] + y*[1 , 2] = [5 , 7]
But I do not know how to show this mathematically not just with a counter example, can someone give me a hint or tell me how can I achieve that?
Note that A, B are elements of R^(nxn)