dimension and rank misunderstood I read the following lemma :  
If $A$ is the matrix of the coefficients with size $m \times n,\,$ then the dimension of solutions of this homogenous system has dimension $n-\operatorname{rank}(A).$
What does "the dimension of solutions " mean? 
Can you give an easy example?
 A: Write the homogeneous system on its matrix form
$$Ax=0$$
hence a solution $x$ is a vector in the kernel of $A$ (or of the linear transformation $f:\Bbb R^n\to\Bbb R^m,\; x\mapsto Ax$) which its dimension by the rank-nullity theorem is $n-\operatorname{rk}(A)$ where $n$ is the dimension of  $\Bbb R^n$.
A: a homogeneous system of equations $Ax = 0$ has always the solution $x=0.$ but it may also have many other solutions. for examplle if you look at $x + y + z= 0$ 
then this system of one equation in three variable has infinitely many solutions. here are some:
$x = -1, y = 1, z = 0: x = -1, y = 0, z = 1$
you can think of many others. in fact $x = -t - s, y = t, z = s$ is always a solution for any values of $s, t$ because they all add up to zero. here you see that this is two fold infinitely many solutions. the number $2$ is called the dimension of the solutions of the homogeneous system of equation $Ax = 0$
A: It refers to $\dim\{x\in R^n:A x=0\}$ (It's easy to see that such a set is a subspace).
