MathDB

4

Part of 1997 IMC

Problems(2)

Infinite product

Source: IMC 1997 day 1 problem 4

10/1/2005
Let α\alpha be a real number, 1<α<21<\alpha<2. (a) Show that α\alpha can uniquely be represented as the infinte product α=(1+1n1)(1+1n2)⋯ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots with nin_i positive integers satisfying ni2≤ni+1n_i^2\le n_{i+1}. (b) Show that α∈Q\alpha\in\mathbb{Q} iff from some kk onwards we have nk+1=nk2n_{k+1}=n_k^2.
real analysisreal analysis unsolved
space mapping

Source: IMC 1997 day 2 problem 4

10/22/2005
(a) Let f:Rn×n→Rf: \mathbb{R}^{n\times n}\rightarrow\mathbb{R} be a linear mapping. Prove that ∃!C∈Rn×n\exists ! C\in\mathbb{R}^{n\times n} such that f(A)=Tr(AC),∀A∈Rn×nf(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}. (b) Suppose in addtion that ∀A,B∈Rn×n:f(AB)=f(BA)\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA). Prove that ∃λ∈R:f(A)=λTr(A)\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)
linear algebralinear algebra unsolved