Linear Algebra Marathon & Questions
This is a marathon thread for linear algebra. Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute.
(mod edit 7/6/17 by dan964)
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To accompany the corresponding calculus thread.
First question (spectral theorem, familiarity with dot product recommended):
Last edited by dan964; 7 Jun 2017 at 4:41 PM.
Last edited by InteGrand; 30 Oct 2016 at 5:34 AM.
A nice and simple question for first year uni students:
Prove the Cauchy-Schwarz inequality.
This is not a first year level problem...
unless you're asking for the vector form of the inequality.
For the sum form of the inequality, the proof is trivial.
Consider the following quadratic equation in x:
In future, you should not omit detail, unless the context makes it clear. Which for this one, probably means vector form.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
(This result gives the first glimpse of why analysis of infinite dimensional vector spaces is far more subtle than that on finite dimensional vector spaces. A sequence can converge with respect to one norm but not with respect to another.)
Last edited by seanieg89; 16 Nov 2016 at 3:17 PM.
If you managed to prove the Cayley-Hamilton theorem for complex matrices using a property of complex matrices (like that diagonalisable complex matrices form a dense set), would this automatically also imply the theorem for any field (or commutative ring), because the theorem is basically a bunch of algebraic identities that hold for any given matrix, and multiplication and addition etc. will all behave the same way regardless of what field the entries of the matrix come from?
Would probably depend on exactly how you proved it. Most proofs would be valid for arbitrary fields, but if you did it using some very special facts about C, I don't know if it is any easier to pass from Cayley Hamilton for C to Cayley Hamilton for F than it is to just prove Cayley Hamilton for F from scratch.
Ill take a good look at this thread along with the advanced marthons while im overseas goodluck people
I understood what you were asking, but the arbitrary field proofs of CH that I know of are rather short, so it seemed an odd detour to go via the complex special case.
In any case, I think you can make your idea rigorous but it takes a bit more care and precision than in your paragraph. I think something like my argument below would work, but forgive any sloppiness as my algebra is quite rusty.
Eg we would first observe that for an arbitrary matrix A over an arbitrary field K, the entries of p_A(A) are all fixed integer polynomials in the entries of A. We can map these integer polynomials to polynomials with coefficients in K by mapping the coefficients to K via the homomorphism m |-> 1+1+...+1 (m ones). The entries of p_A(A) are then given by these image polynomials evaluated at the entries of A, which are in K.
Note however that this polynomial homomorphism is not injective, because 2x and 0 both get regarded as the zero polynomial in (Z/2Z)[X] for instance.
However, in the complex case (in fact in arbitrary characteristic zero case), this mapping is injective. Hence, if the matrix has entries in C, then P_A(A) has entries that are integer polynomials in the entries of A, and if these polynomials vanish for all choices of entries, this implies they have all coefficients equal to zero. By injectivity, this implies that the the original integer polynomials have all coefficients zero and hence the entries of p_A(A) will be zero for arbitrary K.
Prove the following, for any positive integer n:
good luck. you'll need it.
Last edited by Paradoxica; 22 Nov 2016 at 11:03 PM.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
First year courses have different content at different universities, and lecturers who write exams for these courses are way less constrained by things like syllabi than the people who write HSC exams.
Think of this thread as a place to post questions accessible by a first year student (on the knowledge front), with any terminology that would not be standard in all first year courses clarified.
In this sense it is similar to the advanced mx2 marathon, just you are a little less handcuffed re: the methods you can use and a higher standard of rigour is expected. And as always for these threads, I think the ideal question is more demanding on the ingenuity/creativity/intuition front than on the knowledge front.
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