MathDB

vector spaces problem from 1st Greek MO - 1985

Source: 1985 Greece MO Grade XII p4

September 6, 2024

vectorfunctionlinear algebra

Problem Statement

Given the vector spaces

V,W

with coefficients over a field

K

and function

\phi :V\to W

satisfying the relation :

\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)

for all

x,y \in V, \lambda \in K

. Such a function is called linear.
Let

L\varphi=\{x\in V/\varphi(x)=0\}

, and

M=\varphi(V)

, prove that :
(i)

L\varphi

is subspace of

V

and

M

is subspace of

W

(ii)

L\varphi={O}

iff

\varphi

is

1-1

(iii) Dimension of

V

equals to dimension of

L\varphi

plus dimension of

M

(iv) If

\theta : \mathbb{R}^3\to\mathbb{R}^3

with

\theta(x,y,z)=(2x-z,x-y,x-3y+z)

, prove that

\theta

is linear function . Find

L\theta=\{x\in {R}^3/\theta(x)=0\}

and dimension of

M=\theta({R}^3)

.

Back to Problems View on AoPS