Category:Definitions/Rank of Matrix

This category contains definitions related to Rank of Matrix.
Related results can be found in Category:Rank of 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 $\mathbf A$.