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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...

Re: First Year Linear Algebra Marathon $One more for good measure. Suppose $\|\cdot\|_1, \|\cdot\|_2$ are norms on a finite dimensional complex vector space $V$. Prove that there exist positive constants $A,B$ such that $A\|x\|_1\leq \|x\|_2\leq B\|x\|_1$ for all $x\in V$.\\ Prove that the same...

Re: First Year Linear Algebra Marathon $Let $V$ be an arbitrary complex vector space equipped with a norm $\|\cdot\|$. Prove that there exists an inner product on $V$ such that $\langle x ,x \rangle=\|x\|^2$ for all $x\in V$ if and only if the norm satisfies\\ \\...

Re: First Year Linear Algebra Marathon $Let $A$ be an $n\times n$ matrix with coefficients in an arbitrary field $\mathbb{K}$. Explain why $V:=\textrm{Span}(I,A,A^2,A^3,\ldots)$ is a finite dimensional vector space. What is the maximal dimension of $V$? Rigorously justify your answer. $

Re: First Year Linear Algebra Marathon 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 \langle a,b\rangle+\langle b,a\rangle).

Re: Discrete Maths Sem 2 2016 $First we observe that $\phi$ is well defined, because if $x\sim y$, then $f(x)=\exp(ix)=\exp(i(y+2\pi k))=\exp(iy)\exp(2\pi k i)$ which established that $\phi(x)$ is independent of our choice of representative for the equivalence class $x$.\\ \\ Next we observe...

You wrote n^(-n-1/n) in your original post and all subsequent posts until this wolfram comment, in which you have used n^(-1-1/n). The absolute convergence of the series paradoxica mentioned implies the absolute convergence of the former series you did, and yes I am 100% certain of this fact...

Its way more natural IMO. The sequence decreases so rapidly that the sign of terms is irrelevant (i.e. it is absolutely convergent contrary to your original claim). Bounding it above by the sequence n^(-n) makes this abundantly clear. The alternating series test is more useful for things that...

What don't you find appealing about that idea? It is a fundamental skill in analysis to isolate the part of of an ugly expression that dominates behaviour (in this case n^{-n}), and to bound uglier expressions by nicer expressions.

Re: University Statistics Discussion Marathon X\sim \textrm{Bin}(n,p), Y\sim \textrm{Bin}(m,p)\\ \\ f_{X+Y}=\sum_{j=0}^k f_X(j)f_Y(k-j)\\ \\ = p^k(1-p)^{m+n-k}\sum_{j=0}^{k}\binom{n}{j}\binom{m}{k-j}\\ \\=\binom{m+n}{k}p^k(1-p)^{m+n-k}\\ \\ \Rightarrow X+Y\sim \textrm{Bin}(m+n,p).

Claim: A x (B u C) = (A x B) u (A x C) If (x,y) is in the LHS then by definition x is in A and y is in either B or C. If y is in B, then by definition (x,y) is in AxB. Similarly if y is in C then (x,y) is in AxC. Hence LHS c RHS. If (x,y) is in the RHS then either x is in A and y is in B, or x...

From the wording, they probably just wanted you do decide whether: a) Neither of them converge. or b) At least one of them converges. which tests students knowledge of the alternating series test whilst also leaving the red herring of the reciprocal prime series (less subject to elementary...

Using p_n for pi(n) is horrific notation, but it remains true that the first one diverges and the second one converges, and it remains true that proving divergence is nontrivial (quite similar though).

Yes, this is correct. What did you compare the first series to though?

Re: MATH1231/1241/1251 SOS Thread If HSC maths questions (eg irrationality of e or pi) can involve proof by contradiction, why is it so obscene that questions from a first year uni course should? I am sure lots of high school students aren't formally taught proof by contradiction either, but...

Re: MATH1231/1241/1251 SOS Thread They aren't "screwed", but if they don't learn how to prove things (which is completely fine if it is a decision they have made) they shouldn't expect full marks in a typical math course. You could still pass and do reasonably well in most year math courses...

Re: MATH1231/1241/1251 SOS Thread If a student taking any tertiary maths course doesn't know how to argue by contradiction (of which contrapositive is a special case), that is 100% on them. It's like not understanding the notion of disproving a non-existence statement by providing a...

Re: MATH1231/1241/1251 SOS Thread Have they explicitly said you cannot argue by contrapositive/contradicition/etc? These are just fundamental methods of proof, not specific knowledge that may or may not be in syllabus. Your lecturer might not specifically mention them but I would be extremely...

Re: Discrete Maths Sem 2 2016 Definitely not a reduction or a heuristic, the rewording has exactly the same mathematical content. An abstraction perhaps. When we count things in combinatorics, we are almost always counting sets or functions between sets. Sometimes the concept behind such a...

Re: Discrete Maths Sem 2 2016 Eg/ Let's compute the number of surjections from X to Y by inclusion exclusion. (This also computes S(k,n) by dividing out the n! factor.) Let A be set of all functions from X to Y. For each element y of Y, let A(y) be the set of all functions from X to Y that...