# Definition:Rank (Linear Algebra)

## Definition

### Linear Transformation

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

Let the image of $\phi$ be finite-dimensional.

Then its dimension is called the **rank of $\phi$** and is denoted $\map \rho \phi$.

### Matrix

### Definition 1

Let $K$ be a field.

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

Then the **rank** of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$.

That is, it is the dimension of the column space of $\mathbf A$.

### Definition 2

Let $K$ be a field.

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

Let $\mathbf A$ be converted to echelon form $\mathbf B$.

Let $\mathbf B$ have exactly $k$ non-zero rows.

Then the **rank** of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is $k$.

### Definition 3

Let $K$ be a field.

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

The **rank** of $\mathbf A$, denoted $\map \rho {\mathbf A}$ is the largest number of elements in a linearly independent set of rows of $\mathbf A$.