Let y be a column vector of M endogenous variables.

The matrix is on the left and column vectors for point coordinates are used.

It follows that a column vector in represents a document.

Using column vectors, we can represent the same result as follows:

Columns of the identity matrix are added as column vectors for these variables.

Row reduction does not affect the linear dependence of the column vectors.

Write as column vectors, each of which has M rows.

These matrices can act either on real or complex column vectors.

Together they span the space of all 5 dimensional column vectors.

Let be the column vector comprising all those coordinates.