Answer
Let the n$\times$n matrix $A$ be invertible.
Then, by Th.5, for each $\mathrm{b}$ in $\mathbb{R}^{n}$, the equation $A\mathrm{x}=\mathrm{b}$ has the unique solution $\mathrm{x}=A^{-1}\mathrm{b}$.
So, for $b=0$, there is only one solution, $x=0$ (the trivial solution).\\\\
If $A\mathrm{x}=0$ has only the trivial solution, its columns are linearly independent
(see "Linear Independence of Matrix Columns", (3) in section 1.7)
Work Step by Step
Answer contains the proof of the statement.
