Basic columns

Let $$\mathbf E$$ be the echelon form of a matrix $$\mathbf A$$ as below:

$$ \begin{bmatrix} \circledast & \ast & \ast & \ast & \ast \\ 0 &\circledast &\ast &\ast& \ast \\ 0&0&0&\circledast & \ast \\ 0&0&0&0&0& \end{bmatrix} $$

Then, the columns that contain the pivots (i.e., the ones with $$\circledast$$ are called the basic columns. The set of basic columns is linearly independent and also the span of the basic columns is the same as the span of the matrix. Therefore, clearly, all non-basic columns of the matrix can be expressed as a linear combination of the basic columns.

An important feature of basic columns is that the linear combination that recovers non-basic columns through basic columns in the original matrix $$\mathbf A$$ works also on the echelon form $$\mathbf E$$ (see p49 ).