This might seem like a stupid question but would the null space of a matrix and its, say Gaussian elimination transforms, have the same null space. I guess, I am asking if this is valid:(adsbygoogle = window.adsbygoogle || []).push({});

Letxbe in N(A). LetA[itex]_{m}[/itex] be some iteration ofAthrough elimination matrices, i.e.A[itex]_{m}[/itex] =E[itex]_{1}[/itex]E[itex]_{2}[/itex]...E[itex]_{m}[/itex]A. Is N(A) = N(A[itex]_{m}[/itex])?

Seems like an obvious answer with a sorta obvious proof involving expandingA[itex]_{m}[/itex] to the elimination matrices multiplied by the originalAand showing that sincexis inA's nullspace, you just have elimination matrices being multiplied by the zero vector. Is this correct? And if it is, can it apply to other decompositions such as QR via Householder? Can it apply to any arbitrary iterate, 1st, 2nd, etc.?

Thanks for any input.

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# Null Space of a Matrix and Its Iterates

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