Prove if true otherwise give a counter example when it is fa

Prove if true otherwise give a counter example when it is false the set W = {(x_1,3,x_3): x_1 and x_3 are real numbers} is a subspace of R^3 with the standard operations.

Solution

W is not a sub space of R³

Since ( x₁,3 ,x₃ ) , (y₁, 3, y₃) are in W

But ( x₁,3 ,x₃ ) + (y₁, 3, y₃) = ( x₁₊ y₁, 6 , x₃ + y₃) is not in W