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Linear Algebra Isotropic Subspaces

fkhan100

Stumped on the last part of a question which is above.

Any help is appreciated.

Reply 1

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Reply 2

Smaug123

Original post by fkhan100

Stumped on the last part of a question which is above.

Any help is appreciated.

That whole paper was an absolute train-wreck for me :frown:

I would also be interested in seeing this answer! (For completeness, it's question 10 here.)

Reply 3

DFranklin

I'm going to assume you have no problem finding an isotropic subspace of dimension

(n - |\sigma|) / 2

, and you're problem is showing it can't be bigger.

Since B is non-degenerate, we can find an integer k and basis v_1, ...v_k, v_k+1, ..., v_n s,t, the matrix of B w.r.t. this basis is diagonal with the first k entries 1 and the remaining n-k entries -1. (You probably already did this in earlier parts of the question, but as I'm just doing the last bit I need to set up the notation).

Now take v in V'. Then in particular we have B(v, v) = 0. We can of course write

v = \sum \lambda_i v_i

for some \lambda_i. So define a linear map P from V' to span(v_1, ..., v_k) by

P(v) = \sum_1^k \lambda_i v_i

.
Then if P(v) = 0, we must have \lambda_1 = ... = \lambda_k = 0. But then

B(v, v) = -\sum_{k+1}^n \lambda_i ^2

and so all these \lambda_i are 0 too.

So ker(P) = 0, so dim(V') <= k.
A similar argument says that also dim(V') <= n - k.

Since

\sigma = 2k - n

a bit of algebraic manipulation shows no isotropic subspace can have dimension more than

(n - |\sigma|) / 2

(edited 9 years ago)

Reply 4

fkhan100

Original post by Smaug123

That whole paper was an absolute train-wreck for me

Yep that was a tricky paper. You did it as a mock I take it? How many questions did you end up completing? Having only one (or two max) questions per paper on each course makes things a bit more tricky :tongue:

Reply 5

Smaug123

Original post by fkhan100

I got an alpha, possibly two, and maybe three betas. The rest of that year was disconcertingly easy to do as a first mock - I actually got quite worried because I was doing so well on them :P

Reply 6

ElysÇum

Original post by DFranklin

...

Just out of interest, if you don't mind, how come you can do/remember all of this after so many years of having done it last? I find that really impressive!

Reply 7

DFranklin

Original post by Elysǐum

Just out of interest, if you don't mind, how come you can do/remember all of this after so many years of having done it last? I find that really impressive!

Confession time: I was frustrated on this one, because I'd say it was actually a fairly "standard" question when I did the IB Tripos, and I couldn't remember how to do it. So I googled it! The solution I found was a bit vague but had the critical step (projection map from the isotropic subspace to the "+ve" span space), I basically took that bit and reworked the rest.