Prove that in a real vector space V c(alpha - beta ) = c*alpha - c*beta  where c in RR ; alpha,beta in V ?

Oct 2, 2017

See the explanation below

Explanation:

The $2$ operations allowed in a vector space are addition and scalar multiplication. They are called the standard operations on $V$

$\forall u \in V , \exists - v \in V$, this is called the additive inverse of $u$

Here, we have

$\alpha , \beta \in V$ and $c \in \mathbb{R}$

Therefore,

$c \left(\alpha - \beta\right) = c \left(\alpha + \left(- \beta\right)\right)$

Removing the parenthesis

$c \left(\alpha - \beta\right) = c \left(\alpha + \left(- \beta\right)\right) = c . \alpha + c . \left(- \beta\right)$

$= c . \alpha - c . \beta$