# Linear Algebra Field & Vector Space Problem

## Homework Statement

Let V1 and V2 be vector spaces over the same field F.
Let V = V1 X V2 = {f(v1, v2) : v1 $\in$ V1; v2 $\in$ V2}, and de fine addition and scalar multiplication as follows.
 For (v1, v2) and (u1, u2) elements of V , defi ne (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2).
 For (v1, v2) element of V and c $\in$ F, defi ne c  (v1, v2) = (c  v1, c  v2).
a) In the de finitions of addition and scalar multiplication there are three "+" and three "." To
which vector space is each one associated with?
b) Show that V is a vector space. NB: you must provide some reason why each of the axioms is
satis ed.

## Homework Equations

To be absolute honest i have no idea what it means when it asked which vector space it belong to in part a).
ANd for part 2, i do not know where to start.

## The Attempt at a Solution

I know that to proof fields or vector spaces, it has to satisfy with the axioms

Zero vector
addition
scalar multiplication
and etc.
Just have trouble starting this problem

Thanks.

## Answers and Replies

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
Take, for example, ##\vec{x}+\vec{y}## where ##\vec{x},\vec{y} \in \mathbb{R}^2##. You write ##\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}## and ##\vec{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}##. Then
$$\vec{x}+\vec{y} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1+y_1 \\ x_2+y_2 \end{bmatrix}.$$ The plus sign between ##\vec{x}## and ##\vec{y}## represents a different operation than the plus sign between ##x_1## and ##y_1##. Why? Because the first one is about adding two vectors while the second one is about adding two real numbers.