Counterexample (independent columns, no “diagonal form”)

Consider three vectors in $\mathbb{R}^4$. After row reduction you might obtain:

$$\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

• Every column has a pivot.

• The vectors are linearly independent.