Question

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).

Answer 1

what do you need to show in order to show something is a subspace of a vector space?

Answer 2

All I know is that the subset has to satisfy two conditions. That is what I am not sure how to start.

Answer 3

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 \(\lambda w\in S\)

Answer 4

i guess i should have written it with capital letters, but that is the idea

Answer 5

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 \(B+C\in S\)

Answer 6

this should be straight forward because of the distributive law

Answer 7

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

Answer 8

This answers every question I had. Thanks for taking your time.
And then I just realized how simple this should've been.

Answer 9

yw

Answer 10

goal of course is to take the general definition and apply it in the specific case
that is the hard part, rest is routine