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Determine if the following sets are subspaces of $\mathbb{R}^{3}$ under the usual addition and scalar multiplication for $\mathbb{R}^{3}$.

1) $W_1$ = $ \{(a_1,a_2,a_3) \in \mathbb{R}^3 : a_1^2 + a_2^2 = 0$}

2) $W_2$ = $Span(\{(a_1,a_2,a_3)\} : a_1^2 + a_2^2 = 0$})

I think both of them are subspaces, specifically because for $W_1$ the only member is the zero vector, which makes it a subspace by definition. This same argument goes for $W_2$ because if the only member of $W_1$ is the zero vector, $Span(W_1)$ is still the zero vector. Is this correct or am I missing something?

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  • $\begingroup$ The elements of W1 have the form (0,0,a) ie is a line $\endgroup$ – Martín Vacas Vignolo Sep 12 '16 at 2:24
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You are correct that $W_1$ is a subspace, but for the wrong reason. The stipulation that $a_1^2 + a_2^2 = 0$ only says that $a_1 = a_2 = 0$. We can still have that $(0, 0, 7) \in W_1$, for example. In fact, you can see that $W_1$ is the set of all points in $\Bbb R^3$ whose first two components are zero, so its not hard to see this set is closed under linear combinations.

Your second set $W_2$ is also a subspace since you can show that $W_2 = W_1$. Again, neither set is the zero subspace.

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