Linear Algebra Marathon & Questions (1 Viewer)

seanieg89

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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):




 
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InteGrand

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Re: First Year Linear Algebra Marathon

 
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dan964

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Re: First Year Linear Algebra Marathon

A nice and simple question for first year uni students:
Prove the Cauchy-Schwarz inequality.
 

Paradoxica

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Re: First Year Linear Algebra Marathon

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.
 

dan964

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Re: First Year Linear Algebra Marathon

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.
Yes I was asking for inner product spaces, that is for vectors.
 

leehuan

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Re: First Year Linear Algebra Marathon








unless you're asking for the vector form of the inequality.
I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
 
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Paradoxica

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Re: First Year Linear Algebra Marathon










I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
olympiad kids be like that's the second thing I would have thought of...

first is power mean inequality, no less.
 

leehuan

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Re: First Year Linear Algebra Marathon

olympiad kids be like that's the second thing I would have thought of...

first is power mean inequality, no less.
Pretty sure this question is too elementary for olympiad level
 

seanieg89

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Re: First Year Linear Algebra Marathon










I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
Pretty sure you mean min. Also note that it looks mildly messier for complex inner product spaces but the same argument still works. (replace your 2a.b with ).
 

seanieg89

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Re: First Year Linear Algebra Marathon



(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.)
 
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InteGrand

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Re: First Year Linear Algebra Marathon

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?
 

seanieg89

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Re: First Year Linear Algebra Marathon

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.
 

InteGrand

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Re: First Year Linear Algebra Marathon

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.
 

Drsoccerball

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Re: First Year Linear Algebra Marathon

Ill take a good look at this thread along with the advanced marthons while im overseas goodluck people
 

seanieg89

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Re: First Year Linear Algebra Marathon

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.
 

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Re: First Year Linear Algebra Marathon

Prove the following, for any positive integer n:



good luck. you'll need it.
 
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