Long email

I received a long last minute email. I’ll just copy it together with my answers without proper formatting or corrections because I’m also a bit tired.

MK

—————————————————————
Because of lack of time, I will be brief with the answers:

1. Everything is OK except for the statement about the JCF and
the linearly independent eigenvector. A basis for the eigenspace
can be taken as 1. There is just one Jordan block. This should tell you
what the JCF is. (Obviously can’#t be zero).

2. You are right.

3. Right again.

4. The definition as given is correct. For example, might have
f=m. There must have been a misprint in the mark scheme.

5. Yes, these are the same. More precisely, Ker(A)=Ker(-A) for
obvious reasons.

6. One way of thinking about C is as R^2 with some multiplication.
Anyways, the dimension is 2 for the reason you say as well. The dimension
of any field F as a vector space over itself is one. The dimension of
R as a Q-vector space is infinite.

7. See what you can do with L^{98}v.

8. Yes to both questions.

9. This is somewhat tricky. Look at example 125 on the 3704 lecture notes for the general idea.

10. You could, but it might be quicker to take a general symmetric
matrix and show that it can’t work.

11. This is a bit complicated to explain by email in a short amount of
time.

> Dear Prof Kim,
>
> Sorry for the late email. I have quite a few questions on 2201 course
> material. I will be very grateful if you could clear my queries.
>
>
> COURSE SUMMARY AND LECTURE NOTES:
> 1) Summary sheet on course mentions ‘vector space of polynomials with
> complex coefficients of degree at most 5’ Would this be {i, ix, i(x^2),
> i(x^3), i(x^4), i(x^5)} So on D(differentiation):VtoV we get min
> polynomial=ch polynomial=x^5 and so JCF is the zero matrix? And so D has
> only 1 linearly independent eigenvector (i.e. 0)?
>
> 2) Lecture notes page 43: In the proof for diagonalisation theorem, 3
> lines before the end of proof, ‘Hence by the inductive hypothesis, there
> is an orthonormal basis’ shouldn’t this be orthogonal basis?
>
> 3) Page 47 proof of sylvesters law of inertia: In the new basis ci =
> bi/sqrt(q(bi)) and not bi/sqrt(q(bi,bi))?
>
> Also further in the proof ‘q(u)=x1^2+…+xr^2>0’ shouldn’t this be >=0 as
> only then further on U intersection W = {0}
>
> 4) Lecture notes page 25: Definition of minimal polynomial second
> condition on f(T)=0 and f=/0 then deg(f)>=deg(m). Shouldnt deg(f) be
> strictly > deg(m)as suggested in the 2008 mark scheme.
>
> 5) Definition of t-th generalised eigenspace is V(t)(la)=ker((la.Id-T)^t)
> (page27) but in primary decomposition theorem (page28) we use this def as
> V(t)(la)=ker((T-la.Id)^t). Would this not change signs? Are they the same
> because for e.g. kernel implies T(v)=0 so T(-v)=0 too.
>
> 6) What is the dimension of C as an R-vector space? Is this 2 because
> basis is {1, i}?
> What is the basis of C as a C-vector space?
> What is the basis of Q as a Q-vector space?
> What is the basis of R as a Q-vector space?
>
> 7) Suppose there exists a vector v such
> that (L)^100(v)=0,(L)^99(v)=/0. Prove that there exists a
> vector w such that (L)^2(w)=0 and (L)(w)=/0.
> What would be the steps for this proof?
>
>
> HOMEWORK SHEETS:
> 8) Sheet 2 qu7 – Do we only do one long division to get ans?
> Sheet 2 qu9 – Not on syllabus?
>
> 9) How to do sheet 3 qu7?
>
> 10) Sheet 6 qu4 – Can we do this by finding all sym bilinear forms on
> field 2 (8 matrices) and showing none of their quadratic forms are xy?
>
> 11) Sheet 7 qu4 – How do I do part a? Part b is simple. For part c, do we
> explicitly show this for the 9 different possibilities. Most of these
> already in required form (i.e. q00, q01, q10, q11, etc..)
>
>
> Sorry for so many questions. I don’t live near university. Otherwise I
> would have come in to see you. If its easier for you to reply over a
> telephone conversation, please call me on 07971530295.
>
> Thank you very much for your time.
>
>

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