# Thread: Subtraction in vector spaces

1. ## Subtraction in vector spaces

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

2. Originally Posted by 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
$\displaystyle (\lambda+\mu)x=\lambda{x}+\mu{x}$
$\displaystyle (-1)\mu{x}$ 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?
Yes, it is! But why couldn't you just say that $\displaystyle (\lambda-\mu)x=(\lambda+(-\mu))x=\lambda x+(-\mu )x=\lambda x-\mu x$ since you seemed to use all the axioms I did in your proof?