# Showing set is a vector space

Say I have a set of vectors with multiplication and addition both defined. To prove that it is a vector space I have to confirm the eight axioms. When I check the distributive property for scalar multiplication:

$r(u+v) = ru + rv$

(where r is a scalar and u,v are vectors) is the addition here the defined addition or normal addition?

• The former.${}$ – Gerry Myerson Nov 22 '13 at 5:00

It's the vector space addition: if $r$ is a scalar and $u$ is a vector, then $ru$ is a vector (and likewise $rv$), so $ru + rv$ indicates the addition of two vectors. (And so does $r(u + v)$)

On the other hand, if $r$ and $s$ are scalars and $u$ is a vector then the expression

$$(r + s)u$$

indicates addition of two scalars, though if we distribute this $$ru + su$$

then "$+$" once again indicates vector addition.

• Ok. That would make sense. Thanks! – codedude Nov 22 '13 at 5:01

It is the addition as defined/given to you in the problem. There is no such thing as 'normal' addition, even though we are all familiar with certain vector spaces with certain familiar additions, eg $\mathbb{R}$ together with the 'usual' addition everyone learns in primary school. One can easily define a different addition for real numbers for example ( show that the usual rules of addition/axioms are satisfied).

• By normal I meant usual. :) – codedude Nov 22 '13 at 5:02