Question

Let V=R. Define addition and scalar multiplication by a+b=4a+4b. Show whether addition is both commutative and associative.

Answer 1

$a+b=4a+4b\\ c+d=4c+4d\\ c+d+a+b=4a+4b+4c+4d\\ c+d+a+b=4(c+d)+4(a+b)\\$

Let

$A=c+d\\ B=a+b\\ \therefore\\ A+B=4A+4B$

The above defines addition by a+b=4a+4b

Also,

Let s be a scalar

Therefore,

$s(a+b)=4sa+4sb$

The above defines scalar multiplication by a+b=4a+4b

$a+b=4a+4b\\ b+a=4b+4a\\ b+a=a+b$

Hence addition is commutative

$a+b=4a+4b\\ (a+b)+c=4(a+b)+4c=4a+4b+4c\\ a+(b+c)=4a+4(b+c)=4a+4b+4c\\ (a+b)+c=a+(b+c)$

Hence addition is associative