In the lecture notes I'm working through, it says that for a matrix A, the pivot columns of the matrix in reduced row echelon form are a basis for the column space of A. I've also seen some 'proofs' on the internet that support this.
But this doesn't seem correct: take for example $$ \begin{pmatrix} 2 & 1 & 4 \\ 2 & 1 & 4 \\ 2 & 1 & 4 \\ \end{pmatrix} $$
Then in reduced row echelon form this is :$$ \begin{pmatrix} 1 & 0.5 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$
and the pivot column is$$ \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} $$
But there's no way to write one of the columns of the original matrix say
$$ \begin{pmatrix} 1 \\ 1 \\ 1\\ \end{pmatrix} $$ as a linear combination of the pivot column.