What does it mean for the columns of a matrix to be linearly independent?

Question

I've just learnt that det(A) = 0 when the columns of a matrix are linearly dependent, but what does that mean? Could you give me an easy to follow example with numbers please?

Thank you!

Answer 1

The matrix $\begin{bmatrix} 1 & 0 & 5\\ 2 & 4 & 10\\ -5 & 0 & -25 \end{bmatrix}$ has linearly DEPENDENT columns because the last column can be obtained from the first one only by multiplying by $5$. In this case, the determinant of the matrix is $0$ and so the matrix is not invertible.

Answer 2

By definition $\{\vec v_i\}$ are linearly independent if

$$\sum a_i \vec v_i=0 \iff a_i=0 \quad\forall i$$