## Omar91X 2 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. satellite73

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

2. Omar91X

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

3. satellite73

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. satellite73

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

5. satellite73

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. satellite73

this should be straight forward because of the distributive law

7. satellite73

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. Omar91X

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

9. satellite73

yw

10. satellite73

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