MathDB

6

Part of 2012 Pre-Preparation Course Examination

Problems(2)

absolutely convergent sequences

Source: Iran PPCE 2012-Analysis exam-P6

2/14/2012
Suppose that aija_{ij} are real numbers in such a way that for each ii, the series ∑j=1∞aij\sum_{j=1}^{\infty}a_{ij} is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence {bj}j\{b_j\}_j we have lim⁡i→∞∑j=1∞aijbj=0\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0. Prove that
lim⁡i→∞∑j=1∞∣aij∣=0\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0.
limitreal analysisreal analysis unsolved
Inner product vector space

Source: Iran PPCE 2012-Linear Algebra exam-P6

2/16/2012
Suppose that VV is a finite dimensional vector space over the real numbers equipped with an inner product and S:V×V⟶RS:V\times V \longrightarrow \mathbb R is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation T:V⟶VT:V \longrightarrow V such that
∀u,v∈V:S(u,v)=<u,T(v)>\forall u,v \in V: S(u,v)=<u,T(v)>.
We know that there always exists v∈Vv\in V such that W=<v,T(v)>W=<v,T(v)> is invariant under TT. (it means T(W)⊆WT(W)\subseteq W). Prove that if WW is invariant under TT then the following subspace is also invariant under TT:
W⊥={v∈V:∀u∈W<v,u>=0}W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}.
Prove that if dimension of VV is more than 33, then there exist a two dimensional subspace WW of VV such that the volume defined on it by function SS is zero!!!!
(This is the way that we can define a two dimensional volume for each subspace VV. This can be done for volumes of higher dimensions.)
vectorfunctioninvariantlinear algebralinear algebra unsolved