Prove/disprove the following statement:
If b $≠$ 0, then the solution to the matrix: [A|b], mustn't be a plane through the origin.
So far, this seems true to me. I've noticed that if b is not the zero vector, then we arrive at multiple equations, all of which are equal to some value other than 0. This means that they will have intercepts other than zero, thus it can't pass through the origin.
However, I don't know if I am on the right track, or how to prove this.
Any help would be greatly appreciated.
Simple: If the solution is anything through the origin, you'd get $$0=A \cdot 0=b$$ Where $0$ is a properly dimensioned zero-vector
Note that just because there is a nontrivial solution (non-zero), it doesn't directly imply that $0$ is not a solution.