Let VR For uvVuvV and aR define vector addition by uvuv1 and

Let V=R. For u,vVu,vV and aR define vector addition by uv:=u+v1 and scalar multiplication by au:=aua+1. It can be shown that (V,,) is a vector space over the scalar field R. Find the following:

I found the

the sum: 30= 2

the scalar multiple: 63= 13

the zero vector:0_V= 1

what is the additive inverse of x: x= ??

Solution

say a and b are additive inverse of each other then by defintion

a+b=1   (where 1 is zero of the vector space)

a+b-1=1   (using given defintion of sum uv:=u+v1)

a=-b+1+1

a=-b+2

hence additive inverse of u is (-u+2)

we can easily verify this by plugging into uv:=u+v1 where u=u and v=-u+2

uv:=u+v1=u+-u+21=1