- #1

- 97

- 0

## Homework Statement

let [tex]S={(a_1,a_2):a_1,a_2 \in \mathbb{R}}[/tex] For [tex] (a_1,a_2),(b_1,b_2)\in{S}[/tex] and [tex]c\in\mathbb{R}[/tex] define [tex](a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2-b_2)[/tex] and [tex]c(a_1,a_2)=(ca_1,ca_2)[/tex].

show that this is not a vector space

## Homework Equations

vector space axioms

## The Attempt at a Solution

this isn't an exercise in the book, but an example from the book that states that commutativity and associativity of addition and the distributive law all fail, so this in fact is not a vector space

i tried working these out and i think i got commutativity one right

because then you have [tex](a_1+b_1,a_2-b_2)[/tex] does not equal [tex](b_1+a_1,b_2-a_2)[/tex] is this correct?

i got stuck on associativity, i worked it out but to me it seems that it does in fact hold true

haven't check the distributive law though

the book im using is linear algebra by friedberg, insel and spence, second edition