Vectors in a subset I need help making progress on a linear algebra question.
Consider the subset $W = 
\begin{bmatrix}
    x_1 \\
    x_2 \\  
    x_3 \\ 
\end{bmatrix}$
such that $x_1 + x_2 + x_3 = 2$
Does $x = 
\begin{bmatrix}
1 \\
0\\
1\\ 
\end{bmatrix} \in W?$
Does $u = 
\begin{bmatrix}
1\\
1\\
1\\
\end{bmatrix} \in W?$
Does $v = 
\begin{bmatrix}
2\\
1\\
-1\\
\end{bmatrix} \in W?$
My thoughts:
I believe that $x$ and $v$ are in in $W$ because they satisfy the condition. I believe $u$ is not in $W$ because it does not satisfy the condition.
I am not sure whether or not this is correct.
I also need to determine whether or not $W$ is a subspace of $\mathbb{R}^3$ which I do not know how to begin

A vector $(x_1, x_2, x_3)$ would be in $W$ if and only if $x_1+x_2+x_3=2$. So, as you said, $x\in W$, $u\notin W$ and $v\in W$.
Now, check your definition of subspace. Every subspace must contain the zero vector, which is $(0,0,0)$ in this case. But it doesn't satisfy the equation $x_1+x_2+x_3=2$, so $W$ isn't a subspace.
