Making sense of the definition of the rank of a matrix The rank of a matrix being so important, it is frustrating not really knowing what it is, or rather what it represents. The rank of a linear map is the dimension of its range. I don't see what rank of a matrix, which is no function at all, could signify. My textbook, Friedberg's Linear Algebra, gives the following definition:

If $A \in M_{m \times n}(F)$, we define the rank of $A$, denoted $\mathrm{rank}(A$), to be the rank of the linear transformation $L_A : F^n \to F^m$ (left-multiplication transformation)

This of course helps establish some very important results, but I just don't see how the definition 'makes sense' (i.e. compare it to the $\epsilon-\delta$ defintion of a limit which has corresponding intutive sense)
 A: Actually rank gives the dimension of column space and row space.It is the number of linearly independent rows or columns of a matrix.It is also the order of the largest square submatrix with non zero determinant.The best way is to think is through system of linear equations,It shows how many equations you actually have i.e.number of independent equations.
A: Consider a $3\times 4$ matrix $A$
When you multiply $A$ by a column vector $X\in \mathcal {R^4}$ you get a column vector $AX=Y\in \mathcal {R^3}$ thus this matrix defines a linear function $$F:\mathcal {R^4}\to \mathcal{R^3}$$
The range of this function is a subspace of the three dimensional space  $\mathcal{R^3}$ so it can be of dimension $1$, $2$, or $3$ which is called the rank of the matrix $A$
Fortunately the rank of matrix is the number of linearly independent rows or columns of that matrix, that is row rank and column rank are the same as the rank of a given matrix. 
A: If you want a geometric meaning, rank is a number that tells you whether the image $L_A(F^n)$ looks like a point, a line, a plane, something occupying space, or something else.
