# Determining the dimension of the set of solutions of the system $Ax=b$ without solving for $x$ or applying row reduction

Consider the matrix $$A=\begin{bmatrix}2&2&2&4\\1&2&0&-1\\1&3&-1&-4\end{bmatrix}$$

part d) What is the dimension of the solution space of the homogeneous system $$Ax=0$$ and the dimension of the set of solutions (general solution) of the system $$Ax=b$$ (Do not solve for $$x$$ or apply row reduction).

Well, the dimension of the solution space of the homogeneous system $$Ax=0$$ means the dimension of the null space of A, which is $$nullity(A)$$. I found $$rank(A)$$ as $$2$$ in one of the previous parts. $$2 + nullity(A) = 4$$, $$nullity(A) = 2$$. But, what will I do about the general solution?

(No $$b$$ given.)

The dimension of the general solution of $Ax=b$ is the same as the dimension of the
solution of the homogeneous equation $Ax=0$ because the general solution of the equation $Ax=b$ is the sum of the general solution of the equation $Ax=0$ and a special solution of $Ax=b$. If the system is unsolveable, then the dimension is, of course, $0$.