Is a vector apart of a certain Null Space?

Question

I have found that for a certain matrix, the basis for Nul(A) is:

span((-7,2,1,0),(-4,0,0,1)) The question posed is if (-3,2,1-1)^t (ie. transpose) is apart of Nul(A). I have worked out that the vector (-3,2,1,-1) is a linear combination of the two spanning vectors so just wondering if it is that simple?

Answer 1

That vector $(-3,2,1,-1)^\top$ is the difference of the two basis vectors of the kernel (aka $\DeclareMathOperator{Nul}{Nul}\Nul(A)$), thus it is an element of the kernel as well.