Show set of solutions is linear subspace

Question

Here is what I did:

The set of solutions of the above are represented as a column vector with n rows. We prove that it is a subspace first by showing that the set is non empty as there exist the trivial solution. We then justify that it is closed under addition such that adding two column vectors will result in another column vector with the same dimensions $\in \mathbb{R}^n$. Lastly scalar multiple does not change the dimensions, hence the set of solutions is a linear subspace.

Answer 1

HINT

We have that

• $A0=0$
• $A(x_1+x_2)=Ax_1+Ax_2$
• $A(cx)=c(Ax)$