Theorems of solution of a linear system in linear algebra.

Question

Here is a question in GRE subject exam 0568 with its solution, I want to understand why the author said in the solution that "Recall M having linearly independent columns implies the only solution to Mx = 0 is x = 0", could anyone explain this for me, is there a theorem that said this? if so could anyone tell me a book that contains this theorem with its proof ? as I am confused with the theorem given in the following picture, thanks in advance.

Answer 1

Write $M = (v_1,\dots,v_n)$ in terms of its columns. Then $M(x_1,\dots,x_n)^T = x_1v_1 + \dots + x_nv_n$. Saying $Mx$ has no non-zero solution is exactly saying that $v_1,\dots,v_n$ are linearly independent.

If $Mx = Mx'$ and $x \ne x'$ then $M(x - x') = 0$ and $x - x' \ne 0$.

If $Mx = 0$ has no non-zero solutions then $\ker M = \{0\}$ so $\operatorname{im} M = 5$ by Rank-Nullity. Thus $M$ is injective and surjective and hence invertible.

If $M$ is invertible then $\det M \det M^{-1} = \det I = 1$ so $\det M \ne 0$.

If any of this was confusing to you then you should probably go back to a textbook. These one-off answers, I think, won't be much help in comparison.