Question

Let V = R3 and let W = {(x, y, z)  R3| z = x + y}. Prove that W is a subspace of V.

Answer 1

Let V is a linear space and $W\subset V$ is some subset of V. W is a linear subspace in V iff:

1. $\forall(v_1,v_2\in W)(v_1+v_2\in VW;$
2. $\forall(c\in R,v\in W)(c\cdot v\in W);$

We must verify these two properties.

1. Let $v_1=(x_1,y_1,z_1) ,v_2=(x_2,y_2,z_2)\in W$ . It is means that $z_1=x_1+y_1, \space z_2=x_2+y_2$ . Then $v_1+v_2=(x_1+x_2,y_1+y_2,z_1+z_2)\in V=R^3$ be definition of $R^3$ and because of $z_1+z_2=(x_1+y_1)+(x_2+y_2)=(x_1+x_2)+(y_1+y_2)$ we have that $v_1+v_2\in W$ , first condition takes place.
2. Let $v=(x,y,z) \in W, c\in R$ . It is means that z=x+y. Value $c\cdot v=c\cdot(x,y,z)=(cx,cy,cz)\in V=R^3$

by definition? to verify that $cv\in W$ we see that $c\cdot z=c\cdot(x+y)=c\cdot x+c\cdot y$ and this means that the second property is proved also.

Thus we have proved that W is a linear subspace.