anonymous 3 years 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).

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1. anonymous

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

2. anonymous

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

3. anonymous

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 $$\lambda w\in S$$

4. anonymous

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

5. anonymous

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 $$B+C\in S$$

6. anonymous

this should be straight forward because of the distributive law

7. anonymous

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

8. anonymous

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

9. anonymous

yw

10. anonymous

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