Why is this true for matrices? Linearly dependent columns $\implies$ not invertible 
If $A$ is a square matrix with linearly dependent columns, then $A$ is not invertible.

Why is this true for matrices?

If the columns of A are linearly dependent, 
then $a_1\vec{c_1}+\cdots+a_n\vec{c_n}=\vec{0}$ for some scalars $a_1,\cdots, a_n$ (not all 0).
Then $Av=\vec{0}$ where $v=\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}\ne\vec{0}$, so A is not invertible
(since otherwise, multiplying by $A^{-1}$ would give a contradiction).
