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**I-Think** We have to prove some of the basic properties of operations in vector spaces, and sometimes it's easy to make assumptions when doing them, so I just want to make sure I'm correct.

Let V be a vector over a field F

Prove $\displaystyle (\lambda-\mu)x=\lambda{x}-\mu{x}$

Begin with

$\displaystyle (\lambda+\mu)x=\lambda{x}+\mu{x}$

Add the additive inverse

$\displaystyle (-1)\mu{x}$ to both sides

$\displaystyle (\lambda+\mu)x+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})$

Manipulating the L.H.S.

$\displaystyle (\lambda+\mu)x+(-\mu{x})=\lambda{x}+\mu{x}+(-\mu{x})

$

Employing associativity and commutativity

$\displaystyle \lambda{x}+\mu{x}+(-1\mu{x})=\mu{x}+(\lambda{x}+(-1\mu{x})

$

So

$\displaystyle \mu{x}+(\lambda{x}+(-1\mu{x})=\lambda{x}+\mu{x}+(-1\mu{x})

$

Add additive inverse $\displaystyle -\mu{x}$ to both sides again and desired result is obtained.

Is this correct?