I'm just starting to learn linear algebra and I understand Gaussion elimination but I don't understand this proof:
Is the heart of this the fact that all the while when I've been row-reducing matrices to reduced row-echelon form... I've actually been multiplying a matrix by a series of a elementary matrices?
What does the notation $[A\ \ 0]$ mean?
How is the equation $A = E_1^{-1}...E_k^{-1}$ arrived at?