## 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 \[ \begin{pmatrix}a\\b\\c\\ -a-b-c \end{pmatrix} \]