Answer to Question #215639 in Linear Algebra for Jowes

Question #215639

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


1
Expert's answer
2021-07-12T12:24:08-0400

a+b=4a+4bc+d=4c+4dc+d+a+b=4a+4b+4c+4dc+d+a+b=4(c+d)+4(a+b)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+dB=a+bA+B=4A+4BA=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+4sbs(a+b)=4sa+4sb

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

a+b=4a+4bb+a=4b+4ab+a=a+ba+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+4ca+(b+c)=4a+4(b+c)=4a+4b+4c(a+b)+c=a+(b+c)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


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