Chapter 3.5, Problem 31PS

To determine

To Prove:

F must equal G when the m by n matrices A and B have the same four subspaces that are both in row reduced echelon form.

It is proved that $F=G$when there are equal row spaces.

Given:

$A=\left[\begin{array}{l}IF\\ 00\end{array}\right]$and $B=\left[\begin{array}{l}IG\\ 00\end{array}\right]$

Proof:

The matrices $A=\left[\begin{array}{l}IF\\ 00\end{array}\right]$and $B=\left[\begin{array}{l}IG\\ 00\end{array}\right]$are in reduced row echelon form that indicates to consider the rowspace first and not the columnspace.

The columnspace is entirely spanned by the first two columns corresponding to I that indicates that the columnspace is completely independent of F and G.

It is noted that: First row of A $=$ combination of the rows of B

Therefore, as a result of having same rowspace, each row of one matrix is a linear combination of the rows of other matrix which shows $F=G$.

Hence proved.