If A is an m by n matrix, prove that the set of vectors b that are not in C(A) forms a subspace. If A is an m by n matrix, prove that the set of vectors b that are not in C(A)  forms a subspace.
I would like to first understand if I am interpreting the question correctly. My understanding of this question is that I need to first prove that the set of vectors b are equal to the 0 vector, and if b1 and b2 are members of the subspace, then the sum of set b should be a member, and so should some scalar C multiplied by the set of vectors b.  I just don't understand how to actually prove this

Your result is wrong. Maybe this picture can help you figure out.

Since every subspace must contain the zero vector, noted by $\underline{0}$.
We know that $C(A)$ is a subspace for $\mathbb{R}^m$, so $\underline{0}\in C(A) \subseteq \mathbb{R}^m$, that means the zero vector are inside the column space and also $\mathbb{R}^m$.
But if we consider $\mathbb{R}^m\setminus C(A)$, by the picture, it means we are now cancelling the red circle out, so the zero vector is no longer inside this "space".
Therefore, it can not form a subspace.
