# Is this subset also a subspace of $R^3$?

As the title says, I want to determine if this subset H is also a subspace.

$H=\{(x,y,z)\space|\space x=2y^2+3z^3\}\space in\space R^3$

This is homework, so naturally I would prefer explanations over answers.

In order for a set to be a subspace, it needs to pass the addition text and the scalar multiplication test. How can I apply this here?

Hint: pick two random values for $y$ and $z$, and compute the corresponding $x$; that gets you a point $P$ of $H$.

Do it again, to get another point $Q$.

Add the $y$ and $z$ values for $P$ and $Q$ and compute the corresponding $x$; that gives you a point $R$.

Now simply add $P + Q$ to get $R'$. If $H$ is a subspace, then $R'$ must be in $H$. But a point in $H$ that has the $y$ and $z$ coordinates of $R'$ must be $R$. So...does $R = R'$?

If it turns out that it does, do the experiment again, 3 or 4 times. If it always works out, then you can guess that $H$ is a subspace. If it every fails, then $H$ is not.

• Oh my, this solution is simple yet so brilliant! Thanks for the quick response
– Dave
Apr 2, 2017 at 14:27