Description
1. Let S = {(x, y) ∈ R
2
| x + y = 1} be a space defined over the field R with addition
defined as
(a, b) + (c, d) = (a + c, b + d)
and scalar multiplication as x(a, b) = (xa, xb) where x ∈ R and (a, b) ∈ S. Show why
this is not a vector space.
2. Let U = {(x, y) ∈ R
2
| x ≥ 0, y ≥ 0}, with vector addition and scalar multiplication
defined as the previous case. Show why this is not a vector space.
3. Define a set W = R
2 with addition defined as (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
and scalar multiplication defined as c · (x, y) = (cx, y). Show why this is not a vector
space.
4. Let X = {(x, y, z) ∈ R
3
| x+y +z = 0}, with vector addition and scalar multiplication
defined as usual. Show why this is not a vector space.