I know the converse of this statement is true, but is it biconditional?
For $A$ and $B$ $\in M_n(\mathbb{F})$ prove that $Ax=0$ and $Bx=0$ share a common solution set iff we can get $B$ from $A$ through row operation
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linear-algebra
matrices
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2What is the *solution set* of a matrix? – 2017-02-15
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0Yes (assuming they have the same sizes) – 2017-02-15
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0sorry, replace "matrices" with "linear systems" or "augmented matrices" – 2017-02-15
1 Answers
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If the solution spaces of $Ax = 0$ and $Bx = 0$ are the same (that is, if $A$ and $B$ have the same nullspace, then we can indeed get from $A$ to $B$ using row operations.
To show that this is the case: note that for two matrices in reduced row echelon form to have the same nullspace, they must be equal. Then, note that $A$ and $B$ can both be brought to row echelon form (or back from row echelon form) with row operations.