1

Part of 1998 IMC

Problems(2)

real vector space with subspaces

Source: IMC 1998 day 1 problem 1

V

be a 10-dimensional real vector space and

U_1,U_2

two linear subspaces such that

U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6

\varepsilon

be the set of all linear maps

T: V\rightarrow V

T(U_1)\subseteq U_1, T(U_2)\subseteq U_2

. Calculate the dimension of

\varepsilon

. (again, all as real vector spaces)

vectorlinear algebralinear algebra unsolved

IMC 1998 Problem 7

Source: IMC 1998 Day 2 Problem 1

V

is a real vector space and

f, f_{i}: V \rightarrow \mathbb{R}

i = 1, 2, ... , k.

f

is zero at all points for which all of

f_{i }

are zero. Show that

f

is a linear combination of the

f_{i}