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3

Part of 2012 Pre-Preparation Course Examination

Problems(2)

there is no function satisfying the minimum

Source: Iran PPCE 2012-Analysis exam-P3

2/14/2012
Consider the set
A={f∈C1([−1,1]):f(−1)=−1,f(1)=1}\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}.
Prove that there is no function in this function space that gives us the minimum of S=∫−11x2f′(x)2dxS=\int_{-1}^1x^2f'(x)^2dx. What is the infimum of SS for the functions of this space?
functionintegrationSupportreal analysisreal analysis unsolved
Two linear transformations

Source: Iran PPCE 2012-Linear Algebra exam-P3

2/16/2012
Suppose that T,U:V⟶VT,U:V\longrightarrow V are two linear transformations on the vector space VV such that T+UT+U is an invertible transformation. Prove that
TU=UT=0⇔rank⁡T+rank⁡U=nTU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n.
vectorlinear algebramatrixlinear algebra unsolved