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If:

$$W=\left\{\begin{pmatrix} x_1 \\\ x_2 \\\ x_3\end{pmatrix}\in\mathbb{R}^3\ \biggr|\ \ x_1+x_2+x_3=1\right\}$$

...then which of the following statements are TRUE?

Select ALL that apply.

$W$ is not a subspace of $\mathbb{R}^3$ because it doesn't contain the zero vector

$W$ is not a subspace of $\mathbb{R}^3$ because it is not closed under vector addition

$W$ is not a subspace of $\mathbb{R}^3$ because it is not closed with respect to scalar multiplication

$W$ is a subspace of $\mathbb{R}^3$ since it is the solution set of a linear system of equations