Hi, Question 4c of Exam 1 in 18.06SC says "In the vector space M of all 3 by 3 matrices, what subspace S is spanned by all possible row reduced echelon forms R?" The provided answer gives a basis for all upper triangular matrices. Is there an error here? R + R is no longer rref, because the 1's in the pivot positions will become 2's! So this should not be a subspace at all, correct?

Question

Hi, Question 4c of Exam 1 in 18.06SC says "In the vector space M of all 3 by 3 matrices, what subspace S is spanned by all possible row reduced echelon forms R?"  The provided answer gives a basis for all upper triangular matrices.  Is there an error here?  R + R is no longer rref, because the 1's in the pivot positions will become 2's!  So this should not be a subspace at all, correct?

joshdanziger23 · Answer

I agree with you Baffo32! I wonder if it's an error and the examiner meant to ask    about row echelon forms in general and not specifically row reduced echelon forms.

anonymous · Answer

Thanks Josh.  I think you're right.

anonymous · Answer

Hi Baffo and Josh  The question asks for the space "spanned" by echelon matrices //
it will contain many matrices that are linear combinations of echelon matrices but 
as you say they may not be echelon matrices themselves.  All good.  GS

joshdanziger23 · Answer

Thank you Professor! I guess the basic point is that one should read the question! "is spanned by" is not the same as "is made up of"! On a separate matter, can I take the opportunity of your presence on the forum to thank you---and of course MIT---for the wonderful public resource you have created for self studiers? I am hugely appreciative.

Josh

anonymous · Answer

Thanks both of you.  I understand the question now.