Rank
The rank of a matrix is the number of pivots and the number of dimensions that the column space of a matrix occupies.
Rank and Dimension
A matrix with n dimensions exists in Rn space. However, the column space of that same matrix does not necessarily exist in the same number of dimensions.
Consider a matrix like:
┌ ┐ │ 1 2 3│ │ 1 3 4│ │ 1 4 5│ └ ┘
The third column vector can be trivially shown to not be independent; it is a sum of the first and second column vectors. Correspondingly, the eliminated form of the matrix has two pivots and a free variable. As a direct consequence of this being a square matrix (n x n), this also means that there are two pivot rows and a zero row.
┌ ┐ │ [1] 2 3│ │ 0 [1] 1│ │ 0 0 0│ └ ┘
This matrix does not have basis in R3 space. The requirements for basis are that each column be independent and that the matrix span all dimensions.
That matrix does contain two independent columns however, and if those are split off as:
┌ ┐ │ 1 2│ │ 1 3│ │ 1 4│ └ ┘
Then this new matrix has basis for the column space of the matrix, which happens to be 2 dimensional.
Effectively, this matrix expresses a 2-dimensional plane that exists within a 3-dimensional space, but it is bound to that plane and cannot vary across the third dimension.
Categories for Solutions
If a matrix has a pivot in each column, it is said to be full column rank. The only null space is the zero vector (i.e. [0 ...]). Ax = b either has one solution or is not solvable. Incidentally, the reduced row echelon form (R) looks like the identity matrix (I) with some number of zero rows.
If a matrix has a pivot in each row, it is said to be full row rank. This only means that Ax = b can be solved for any b.
If a square matrix has full rank, the reduced row echelon form (R) of the matrix is the identity matrix (I). The only null space is the zero vector; Ax = b can be solved for any b; there is exactly one solution for any given b.
If a matrix is none of the above, Ax = b either can be solved for any b or is not solvable.