Row equivalence. What is it exactly? When matrices are row equival

Irrerbthist6n 2021-12-20 Answered
Row equivalence. What is it exactly?
When matrices are row equivalent... why is this important? If a matrix like:
[1011]
is row equivalent to the identity matrix (add 3 times the first row to the second), what does that mean exactly? Why is this a concept that we have to know as students of linear algebra? These matrices arent
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einfachmoipf
Answered 2021-12-21 Author has 32 answers

[10|031|0]R2R2+3R1 (1)
[10|001|0] (2)
The above corresponding system of homogeneous equations convey the same information.
x = 0 x = 0 −3x + y = 0 y=0
Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
Two matrices in reduced row echelon form have the same row space if and only if they are equal.

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Donald Cheek
Answered 2021-12-22 Author has 41 answers
My reason for why the concept of row equivalence is important is that the solutions to the two matrix equations
Ax=b
and
Bx=b
are the same as long as A and B are row equivalent. Often time, you want to reduce an original metric equation Ax=b to an equation Bx=b that is easier to solve, where B is row equivalent to A since row operations do not change the solution set.

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