How to express the basis of a subspace? Find a basis for this subspace of $\mathbb{R}^4$:
All vectors whose components add to zero.  
I think I know what this is asking but I don't know how to express the final answer.
All vectors whose components add to zero means we have $3$ free variables in $\mathbb{R}^4$: 
$$\begin{bmatrix} a\\ b\\ c\\ -(a+b+c)\\ \end{bmatrix}$$
Since we have three free variables we would need three vectors in a basis, but do you express the final answer as
$$a\begin{bmatrix} 1\\ 0\\ 0\\ -1\\ \end{bmatrix} +b\begin{bmatrix} 0\\ 1\\ 0\\ -1\\ \end{bmatrix} +c\begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
is a basis for all vectors whose components add to zero in $\mathbb{R}^4$?
 A: That is right you have find the basis exactly up to above comment!
the basis is$\{(1,0,0,-1),(0,1,0,-1),(0,0,1,-1)\}.$
A: As far as I know, there is no unique answer. You can say that
$$\begin{bmatrix} 1\\ 0\\ 0\\ -1\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ -1\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
is a basis.
However, you can also say that
$$\begin{bmatrix} a\\ 0\\ 0\\ -a\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ -1\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
$$\begin{bmatrix} 1\\ 0\\ 0\\ -1\\ \end{bmatrix} \begin{bmatrix} 0\\ b\\ 0\\ -b\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
$$\begin{bmatrix} a\\ 0\\ 0\\ -a\\ \end{bmatrix} \begin{bmatrix} 0\\ b\\ 0\\ -b\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$$
are also bases.
A: All vectors whose components add to zero$\iff$Nullspace of the matrix $A=\begin{bmatrix}1&1&1&1\end{bmatrix}$
Solutions to the equation
$$
Ax=0\implies A=\begin{bmatrix}1&1&1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}\\
x_1+x_2+x_3+x_4=0\\
\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=x_2\begin{bmatrix}-1\\1\\0\\0\end{bmatrix}+x_3\begin{bmatrix}-1\\0\\1\\0\end{bmatrix}+x4\begin{bmatrix}-1\\0\\0\\1\end{bmatrix}
$$
All vectors whose components add to zero=$span\bigg\{ \begin{bmatrix}-1\\1\\0\\0\end{bmatrix},\begin{bmatrix}-1\\0\\1\\0\end{bmatrix},  \begin{bmatrix}-1\\0\\0\\1\end{bmatrix}\bigg\}$
