Linear Algebra (Vector)

Question

Determine whether or not the set of vectors identified is a subspace of:

#1.    

 R³ if W is the set of all vectors of the form (x, y, 1) under the usual vector addition and scalar multiplication.

Answer 1

Solution:

The subset W = { (x,y,1) | x, y ∈ R} is not closed under the addition because if we take u = (1,1,1) and v = (0,0,1), then u ∈ W and v ∈ W, but u + v = (1,1,2) ∉ W. Therefore, W is not a subspace of R^3.