1
$\begingroup$

In problem set 2.2, question 28 of the book Linear Algebra and its Applications by Gilbert Strang, we are asked to express a matrix $A$ of rank $r$ as a product of two matrices:

  1. A matrix containing the pivot columns of $A$ and
  2. A matrix containing the first $r$ rows of $R$.

For instance,

$$ \begin{bmatrix}1 & 3 & 3 & 2\\2 & 6 & 9 & 7\\ -1 & -3 & 3 & 4\end{bmatrix} = \begin{bmatrix}1 & 3\\2 & 9\\ -1 & 3 \end{bmatrix} \begin{bmatrix}1 & 3 & 0 & -1\\0 & 0 & 1 & 1\end{bmatrix} $$

Why is this possible and what is the significance of such a representation?

$\endgroup$
  • $\begingroup$ $R $ being the reduced row echelon form of $A $? $\endgroup$ – AnyAD Dec 7 '18 at 12:12
  • $\begingroup$ @AnyAD Yes, it is the row reduced echelon form. $\endgroup$ – Mohideen Imran Khan Dec 7 '18 at 12:24
2
$\begingroup$

The columns that have pivots are the linearly independent columns of $A $. So all other columns of $A $ are linear combinations of these. The matrix with non-zero rows from the reduced form contain the relevant coefficients of linear dependence (for each column that can be expressed as a linear combination of the linearly independent/pivot set). The pivot columns in the matrix when premultiplied by the matrix of pivot columns simply extract the pivot columns.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.