Answer
If $\mathbf{A}\vec{x}=\vec{b}$ has at most one solution for every vector $\vec{b}$ in $\mathbb{R}^{m}$, then $\mathbf{A}\vec{x}=\vec{0}$ must have exactly one solution (i.e., $\vec{x}=\vec{0}$), since every homogeneous linear system has the trivial solution. But $\mathbf{A}\vec{x}=x_{1}\vec{v}_{1}+x_{2}\vec{v}_{2}+...+x_{n}\vec{v}_{n}$, where the vectors $\vec{v}_{i}$ are the columns of $\mathbf{A}$; hence, the vector equation $x_{1}\vec{v}_{1}+x_{2}\vec{v}_{2}+...+x_{n}\vec{v}_{n}=\vec{0}$ has only the solution $x_{1}=x_{2}=...=x_{n}=0$. Therefore, by the definition of linear independence, the columns of $\mathbf{A}$ are linearly independent.
Work Step by Step
The definition of linear independence is given on page 57.
