Consider a 5 x 4 matrix.... We are told that the vector, 1 2 3 4 is in the kernel of A. Write v4 as a linear combination of v1,v2,v3 I'm a bit confused. Since this is a kernel of A, the kernel is a subset of R^m, therefore the other columns are linear combinations and therefore redundant. (since this is the only column represented) So, that means I can have the columns be anything I want, so why can't they just all be the same? Is this too easy or am I missing something?