Question

Linear Algebra: If A is a matrix and x and b are vectors, what is a necessary and sufficient constraint on A to ensure Ax=b has a non-zero solution for x?

Answer 1

WARNING: I AM REALLY TIRED RIGHT NOW. I AM NOT SURE WHETHER THE FOLLOWING IS CORRECT.

It is clearly necessary for \(\underline{b}\) to be non-zero in order for \(\underline{x}\) to be non-zero. If \(\underline{b}=\underline{0}\), then \(\underline{x}=\underline{0}\) will be a solution to this equation.

Let \(A\) be a mxn matrix and suppose that \(b\in V\), it seemed to me that \(\operatorname{rank}(A)=\dim(V)=m\) is both necessary and sufficient for \(A\underline{x}=\underline{b}\) to have a non-zero solution given \(\underline{b}\neq\underline{0}\). If \(\operatorname{rank}(A)<\dim(V)\), it will not span the entire \(V\) and some \(\underline{v}\in V\) will not have at least one \(\underline{x}\) corresponding to it. \(\operatorname{rank}(A)>\dim(V)=m\) seems contradictory to me. If \(n>m\), then \(\operatorname{rank}(A)\leq m\). If \(n<m\), then \(\operatorname{rank}(A)\leq n<m\). If \(n=m\), then clearly \(\operatorname{rank}(A)\leq n=m\). Neither way it will be greater than \(m\).