How do I determine if this is a subset of a subspace?

So I know that for a subset to be a subspace it has to satisfy the following properties:

1. Contain the zero vector
2. Closed under scalar multiplication
3. Closed under addition

I however do not know how to about determining whether:

W = {[x,y,z] for all in R^3 |x-2y+3z = 0}

is a subspace of R^3

• Did you mean "is a subspace" on the last line? Isn't it obvious that $W$ is a subset? Aug 11 '14 at 13:25
• This is a collection of ordered triples of reals, so it is a subset of $\mathbb{R}^3$. Aug 11 '14 at 13:25

3 Answers

Take any $\mathbf{x} = (x,y,z)^T$ in $W$. $\mathbf{x}$ satisfies $x - 2y + 3z = 0$. Now take the vector $a \mathbf{x}$, it satisfies $(ax - 2ay - 3az) = a(x-2y-3z) = a 0 = 0$. So the second condition is satisfied. Now take two vectors $\mathbf{x}_1 =(x_1,y_1,z_1)$ and $\mathbf{x}_2 = (x_2,y_2,z_2)$ in $W$. Both satisfy the equation. As the equation is linear, $c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2$ also satisfies the equation $\forall c_1,c_2 \in \mathbb{R}$. Finally, it is trivial to show that $\mathbf{0} = (0,0,0)$ also satisfies the equation.

One way to do this is consider the function $f:\mathbb{R}^3 \to\mathbb{R}$ given by $$f(x,y,z) := x-2y+3z$$ Show that this function is linear. Now check that for any linear function $f$ as above, the set $$\{(x,y,z)\in \mathbb{R}^3 : f(x,y,z) = 0\}$$ is a subspace of $\mathbb{R}^3$ by checking all the axioms.

This is somewhat more conceptual and will help you solve other such problems as well.

It is rather straight forward, all you have to do is follow the three conditions: 1. $0$ abviousliy belongs to W since : $0-2*0+3*0 = 0$ 2. if you take $(x,y,z)$ in W and multiply by a scalar a you get - $ax-a2y+a3z = a(x-2y+3z) = a*0 = 0$. 3. and for the third if you take $(a,b,c),(d,e,f)$ in W and take their sum you get $(a+d,b+e,c+f) = (a+d)-2(b+e)+3(c+f) = (a+2b-3c)+(e-2d+3f) = 0+0 = 0$ so their sum is in the subspace as well.