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Omar91X
Let A be a fixed vector in R^(nxn) and let S be the set of all matrices that commute with A; that is, S={B | AB=BA} Show that S is a subspace of R^(nxn).
what do you need to show in order to show something is a subspace of a vector space?
All I know is that the subset has to satisfy two conditions. That is what I am not sure how to start.
i think you only need to show two things 1) if \(w, v\in S\) then \(w+v\in S\) and 2) if \(w \in S, \lambda\in \mathbb{R}\) then \(\lambda w\in S\)
i guess i should have written it with capital letters, but that is the idea
so you have two jobs 1) show that if \(B, C\) commute with \(A\), that is if \(AB=BA\) and \(AC=CA\) then \[A(B+C)=(B+C)A\] i.e. show that if \(B\in S\) and \(C\in S\) then \(B+C\in S\)
this should be straight forward because of the distributive law
you also have to show if \(AB=BA\) then \(A\lambda B=\lambda BA\) which again should be straight forward by the definition of scalar multiplication
This answers every question I had. Thanks for taking your time. And then I just realized how simple this should've been.
goal of course is to take the general definition and apply it in the specific case that is the hard part, rest is routine