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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).
 one year ago
 one year ago
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).
 one year ago
 one year ago

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satellite73Best ResponseYou've already chosen the best response.1
what do you need to show in order to show something is a subspace of a vector space?
 one year ago

Omar91XBest ResponseYou've already chosen the best response.0
All I know is that the subset has to satisfy two conditions. That is what I am not sure how to start.
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
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 \(\lambda w\in S\)
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
i guess i should have written it with capital letters, but that is the idea
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
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 \(B+C\in S\)
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
this should be straight forward because of the distributive law
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
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
 one year ago

Omar91XBest ResponseYou've already chosen the best response.0
This answers every question I had. Thanks for taking your time. And then I just realized how simple this should've been.
 one year ago

satellite73Best ResponseYou've already chosen the best response.1
goal of course is to take the general definition and apply it in the specific case that is the hard part, rest is routine
 one year ago
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