Proving Vector Space of 3-Tuples Fulfilling 3x1 - x2 + 5x3 = 0

[alexngo]

Homework Statement

show that the collection of all ordered 3-tupples (x1,x2,x3) whose components satisfy 3x1 - x2 + 5x3 = 0 forms a vector space with the respect the usual operation of R3.

Homework Equations

3x1 - x2 + 5x3

The Attempt at a Solution

we tried it by addition and multipication..solutions would be appreciated asap

 

[tiny-tim]

Welcome to PF!

Hi alexngo! Welcome to PF!

we tried it by addition and multipication..

ok! … show us what you get!

 

[alexngo]

by showing that it respects both addition and multiplication does this proves it to be a vector space or we need to show taht it satisfies all 10 axioms

 

[CompuChip]

Technically you need to show all 10 of them. But note that, if your set is a vector space, then it is a subspace of R^3. In simpler terms: the "vectors" from your vector space are just vectors as you know them. So the axioms about distributivity and associativity carry over to the subspace (for example: if the addition of three arbitrary 3-d vectors is associative, then the addition of three special ones of the form (x, y, (y-3x)/5) is definitely associative as well). In fact there is a reduced set of axioms which you can use to show that a subset of a vector space is a vector space.

 

[tiny-tim]

by showing that it respects both addition and multiplication does this proves it to be a vector space or we need to show taht it satisfies all 10 axioms

You only need the closure axioms …

you don't need to prove eg a + b = b + a because so long as you've proved closure, ie that b + a is in the collection, then a + b = b + a is automatically satisfied.

But of course, you do still need to prove closure under multiplication by a scalar.

EDIT: oooh, CompuChip beat me to it! ​