Dimension of the Solution Space

Suppose a vector space  
\[V\]
  is a subspace of  
\[\mathbb{R}^4\]
  with every vector  
\[\mathbf{v} = \begin{pmatrix}a\\b\\c\\d\end{pmatrix} \in V \]
  satisfying  
\[a+b+c+d=0\]
.
The dimension of the solution space satisfying the above condition is the maximum number of independent vectors with components  
\[a,b,c,d\]
  satisfying  
\[a+b+c+d=0\]

Obviously this is 3.
We can take 
\[a=1,b=c=0,d=-1\]
  as one vector.
We can take 
\[a=0, b=1,c=0,d=-1\]
  as another vector.
We can take 
\[a=0,b=0,c=1,d=-1\]
  as one vector.
None of these three vectors can be expressed as combinations of the other two, and any vector in  
\[V\]
&nbsp: can be written in the form  
\[ \begin{pmatrix}a\\b\\c\\ -a-b-c \end{pmatrix} \]