How to perform a basic full rank factorization?

Question

What is the general process for performing full rank factorization on a matrix?

For example, if I am given $$A = \begin{bmatrix} 2 & 1\\ 4& 2\\ 6 &3 \end{bmatrix}$$

What operations do I perform on this matrix, to get the following solution? $$=\begin{bmatrix} 1\\2 \\3 \end{bmatrix}\begin{bmatrix} 2 &1 \end{bmatrix}=BC$$

I ask, because if I am given a different $A$ matrix, where the factorization isn't so obvious, I would like to know the best method to achieve full-rank factorization!

Answer 1

• Find a basis for the row space
• Write the rows as linear combinations of basis elements

or

• Find a basis for the column space
• Write the columns as linear combinations of basis elements