Would $x_1^2 + x_2^2 = 0$ be considered an equation for a linear subspace of $\mathbb{R}^2$?

Question

I understand the equation satisfies the subspace containing the zero vector however I don't know how to interpret the squares on $x_1$ and $x_2$. Can anybody explain to me the difference it makes, if any, that the terms are squared?

Answer 1

The equation $$ x_1^2+x_2^2=0 $$ has one solution, $x_1=x_2=0$. This is the trivial subspace of $\mathbb{R}^2$.

Answer 2

If the terms were not squared, $x_1 + x_2 = 0$ would define a line $x_1 = -x_2$. However, since the terms are squared, both terms are forced to be $0$, so the subspace is the zero subspace (which is, in fact, a linear subspace).