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
Given:
Proof:
The matrices
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
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
Hence proved.
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Introduction to Linear Algebra, Fifth Edition
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