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