Maximum number of linearly dependent columns of a $m \times n$ matrix Question statement : True or false ? 
If $A$ is a ($4 \times 8$) matrix, then any six columns are linearly dependent. 
Textbook answer : True. 
I can't understand the reasoning behind this. From my perspective , for such matrix there are at most 4 pivot columns, therefore at most 4 linearly independent columns, which would imply there are at most 4 linearly dependent columns ? 
I don't know where to begin, help would be appreciated.
 A: Your attempt is almost there: ...therefore at most 4 linearly independent columns, therefore any six columns must be linearly dependent.

The proper way to state this is that the rank of an $n\times m$ matrix is at most $\min(m,n)$, therefore your matrix has rank at most $4$, therefore any $5$ or more rows (respectively, columns) must be linearly dependent.
A: 
From my perspective , for such matrix there are at most 4 pivot columns, therefore at most 4 linearly independent columns, which would imply there are at most 4 linearly dependent columns ?

You are right about the maximum number of pivot columns and therefore the maximum number of linearly independent columns, but what makes you think the italic statement follows from that?
Take a look at:
$$\begin{pmatrix}
\color{blue}{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \color{blue}{1} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \color{blue}{1} & \color{red}{2} & \color{red}{3} & \color{red}{4} & \color{red}{5} & \color{red}{6} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}$$


*

*How many pivot columns are there?

*What is the maximum number of linearly independent columns you can pick?

*What is the maximum number of linearly dependent columns you can pick?

