4

Part of 2012 Pre-Preparation Course Examination

Problems(2)

each compact set has a compact pre-image

Source: Iran PPCE 2012-Analysis exam-P4

X

Y

are metric spaces and

f:X \longrightarrow Y

is a continious function. Also

f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R

f_1(x,t)=(f(x),t)

x\in X

t\in \mathbb R

is a closed function. Prove that for every compact set

K\subseteq Y

, its pre-image

f^{pre}(K)

is a compact set in

X

topologyreal analysisreal analysis unsolved

Upper triangular matrix

Source: Iran PPCE 2012- Linear Algebra exam-P4

Prove that these two statements are equivalent for an

n

dimensional vector space

V

:

\cdot

For the linear transformation

T:V\longrightarrow V

there exists a base for

V

such that the representation of

T

in that base is an upper triangular matrix.

\cdot

There exist subspaces

\{0\}\subsetneq V_1 \subsetneq ...\subsetneq V_{n-1}\subsetneq V

such that for all

i

T(V_i)\subseteq V_i

linear algebramatrixvectorlinear algebra unsolved