Linear dependency of a Matrix's columns and its relation with Matrix product

Question

With regard to this answer, I would like to know how come that if the product of a square matrix $M \in R_{n,n}$ with a non-null column vector $X \in R_{n,1}$ is $null$ then $M$'s columns or rows are linearly dependent. In a more formal way :

$M=(V_{1}, V_{2}, ..,V_{n}) \enspace and \enspace M.X=0 \enspace and\enspace X\neq0 \Rightarrow \exists \enspace V_{i} \enspace such\enspace as \enspace V_{i}=\sum_{k=1}^{n} a_{k}.V_{k}$

Could someone please elaborate more on why this is true ? A reliable reference will be more than appreciated.

Answer 1

Suppose the rows of your matrix $M$ are are linearly independent.

Then the matrix $M$ is invertible.

That is, we have a matrix $M^{-1}$ such that $$M^{-1}M=I$$

Now if $MV=0$ for some $V\ne 0$, then $$ V= IV=(M^{-1}M)V=M^{-1}(MV)=0$$ which is a contradiction.

Thus if $MV=0$ for some $V\ne 0$ then the rows of $M$ are linearly dependent.

Answer 2

If $X = (x_1,x_2, \dots, x_n)^T,$ then you can check that $MX = x_1V_1+x_2V_2+\dots+x_nV_n,$ so the statement is just the definition of linear independence.