Is (a,b,c) where b=a+c a Subspace of r^3?

[kwal0203]

Homework Statement

Determine whether the following is a Subspace of r^3:

All vectors of the form (a,b,c), where b=a+c

The Attempt at a Solution

The answer in the book says it is not a subspace but I can only find examples that show it is a Subspace I.e.

Let u=(a,a+c,c)=(1,2,1), v=(a,a+c,c)=(2,4,2) and k=2 then

U+v=(3,6,3)=(a,a+c,c) so it's closed under addition

Ku=2(a,a+c,c)=2(1,2,1)=(2,4,2)=(a,a+c,c) so it is closed under scalar multiplication

Maybe I don't understand the concept correctly am I doing something wrong?

 

[HallsofIvy]

If your textbook says this is not a subspace, it is wrong. It is, exactly as you say, the subspace of R³ of all vectors of the form (a, b, c)= (a, a+ c, c)= (a, a, 0)+ (0, c, c)= a(1, 1, 0)+ c(0, 1, 1). In other words, it is the two dimensional subspace with (1, 1, 0) and (0, 1, 1) as basis.