Exploring Right Ideals in M(n,n)

[Gale]

"right ideals"

Ok, so this is an extra credit question on a test, i haven't really tried it yet, but the test is thurs, so i figured i'd try to post this to see what anyone says, and then see what i work out, or whatever. I don't even know what "right ideals" means, but our prof said that's what the question was about... so i figured... ya...

Let R be a subspace of V = M(n,n) such that AB is in R whenever A is in R. Let W be the subspace of R_n spanned by all AX with A in R, X in R_n.

A) show that for any matrix A in M
i) Aej = Aj and ii) A= summation(AjeJ)

B) show that if AX is in W for every X in R_n then Aj is in W.

C) Write Aj as a linear combination of products A(ij)X(ij), A(ij) in R, X(ij) in R_n

D) use ii) to show that if A is as in B), then A is in R
E) show that R consists of all matrices A in M with AX in W for all X in R_n

OOOOK... so that's the problem. The only hint he gave us was the "right ideals" thing. So i'll google that and see if i can work this all out in the morning. I pretty much have tons of trouble with this stuff though, so we'll see. Any help would be totally awesome! oh, and if you don't understand the question... join the club... i can try to explain his notation if its weird... but that's about it. thanks in advance...

 

[Hurkyl]

I'm boggled by most of the notation -- you probably should explain what all of it means. (The only thing I'm sure of is that M(n, n), or $M_{n, n}$ means the vector space of nxn matrices, presumably with real entries!)


I can make a comment that I think will be useful -- if you write the identity matrix, I as a sum of other matrices, then it is sometimes productive to observe that A = AI, and then replace I with that sum... and then later use this "decomposition" by substuting it in for A.

 

[Gale]

mk, i'll try and rewrite in latex, even though our test is written entirely this way, i'll do my best to explain...

Let R be a subspace of $$V = M_{n,n}$$ such that AB is in R whenever A is in R. Let W be the subspace of $$R_{n}$$ spanned by all AX with A in R, X in $$R_{n}.$$

a)show that for any matrix A in M
i) Aej = Aj and ii) $$A= \Sigma (Ajej)$$

[ok, so here, ej means the the j-ith column of the identity matrix... i think... see if that makes sense. Aj is the jth column of A i guess, and the second ej could be the jth row of the identity or jth column... I'm not sure]

b) ...
[same as before, i guess R_n means $$R_{n}$$ if that helps... Aj means jth column of j i assume...]

c) again.. tex doesn't change anything...
[A(ij) is the entry of A in the ith row and jth column. X(ij) is the entry in X in the ith row jth column.]

d) and e) stay the same, i don't know how i could explain those any better...

Anyways, ya, he's notation is sort of weird i guess... but i don't have a book to really notice the difference anyways. So, maybe that helps... again, i'll do out some of the work if i can in the morning. Its way to late to make sense of this right now.

 

[Hurkyl]

I don't know what R_n means, though.

C) is odd -- Aj is a matrix, but if A_ij and X_ij are scalars... no linear combination of scalars can be a matrix. My best guess is he's again doing something weird... instead of A_ij being the (i,j)-th entry of A, he's saying that A_ij is just yet another matrix. (because he says A(ij) in R) I guess that's why he wrote A(ij) instead of A_ij?

 

[Gale]

R_n just means numbers in the nth dimension.

 

[snoble]

I'm not so sure that X(ij) is a scaler. I think that X(ij) is just the vector that goes with the scaler A(ij). The question does say that X(ij) is in R_n which suggests we are talking about a vector.

A right ideal (in the case of vectors) is exactly the definition of R. ie a vector subspace of a ring V(where you consider the ring a vector space over its centre) with the property that if $$A\in R$$ and $$B\in R$$ then $$AB \in R$$.

The only thing to watch out for from parts A through D is that you show some care to recognize when R means the right ideal and when R means the reals. Part A section ii is a little bit of a mystery to me, even if you take ej to be a row vector.

B is simple. You just need to consider the right X in R_n. Hint you use these vectors all the time. You've even used them in part A

Again in C you just need to choose the right vector for X(ij)

D is a little trickier. I might be finding it tricky because I don't understand what ii) means. First can you show Aj is in W? Then consider what it means to be in W (ie the span of some set). With that in mind can you right out Aj as a sum of elements of the form Bx with B in right ideal and x some vector. Now can you right out A as a sum of Matrices of the form BX with B in the right ideal and X just some matrix. What does that mean about A?

And then E becomes easy again. If A is in R does it satisfy the condition? And you showed in D that if A satisfies the condition then A is in R.

I don't think I've given too much away and I hope I've cleared up the problem a little.

Steven