# What does it mean for the columns of a matrix to be linearly independent? [closed]

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!

• The identity matrix has determinant $1$. Linear independence of the columns means that the column vectors of the matrix are linearly independent. You should review this definition (e.g. google "linear independence").
– Dave
Mar 30, 2018 at 23:01
• Oh sorry, honest mistake. Ok thanks
– Jess
Mar 30, 2018 at 23:01
• You mean the determinant is $0$ when the column are linearly dependent that is not linearly independent. Mar 30, 2018 at 23:17
• @Jess Please remember that you can choose an answer among the given if the OP is solved, more details here meta.stackexchange.com/questions/5234/…
– user
Apr 1, 2018 at 8:53

By definition $\{\vec v_i\}$ are linearly independent if
$$\sum a_i \vec v_i=0 \iff a_i=0 \quad\forall i$$
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.