\[V\]
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
\[V\]
: can be written in the form \[ \begin{pmatrix}a\\b\\c\\ -a-b-c \end{pmatrix} \]