Definition:Rank (Linear Algebra)

Linear Transformation
Let $$\phi$$ be a linear transformation from one vector space to another.

If the range of $$\phi$$ is finite-dimensional, its dimension is called the rank of $$\phi$$ and is denoted $$\rho \left({\phi}\right)$$.

Matrix
Let $$K$$ be a field.

Let $$\mathbf{A}$$ be an $m \times n$ matrix over $$K$$.

Then the rank of $$\mathbf{A}$$, denoted $$\rho \left({\mathbf{A}}\right)$$, is the dimension of the subspace of $$K^m$$ generated by the columns of $$\mathbf{A}$$.