Non-square matrix and linear independence

[torquerotates]

Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.

 

[rasmhop]

Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.

Yes. For instance,
$$\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right]$$
Of course it will have to have more rows than columns.

 

[HallsofIvy]

If, on the other hand, the matrix has more columns than rows, the columns cannot be independent. If there are say, m columns and n rows, with m> n, then the columns are n dimensional vectors and a set of m vectors of n dimensions, with m> n, cannot be independent.