Matrix/Column rank/Row rank/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}M$ denote an ${}m\times n$-matrix over ${}K$. Then the dimension of the linear subspace of ${}K^{m}$, generated by the columns, is called the column rank of the matrix, written

$\operatorname {rk} \,M.$

Lemma

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote ${}K$-vector spaces of dimensions ${}n$ and ${}m$. Let

$\varphi \colon V\longrightarrow W$

be a linear mapping, which is described by the matrix ${}M\in \operatorname {Mat} _{m\times n}(K)$, with respect to bases of the spaces. Then

${}\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,$

holds.

Proof

See exercise.

$\Box$

To formulate the next statement, we introduce row rank of an ${}m\times n$-matrix to be the dimension of the linear subspace of ${}K^{n}$ generated by the rows.

Lemma

Let ${}K$ be a field, and let ${}M$ denote an ${}m\times n$-matrix over ${}K$. Then the column rank coincides with the row rank. The rank equals the number ${}r$ from

fact.

Proof

In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. The row rank of ${}M$ equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank ${}r$, since the first ${}r$ rows are linearly independent, and, apart from this, this, there are only zero rows. It has also column rank ${}r$, since the ${}r$ columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these ${}r$ columns. By exercise, the column rank is preserved by elementary row manipulations.

$\Box$

Both ranks coincide, so we only talk about the rank of a matrix.

Corollary

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix

over ${}K$. Then the following statements are equivalent.

1. ${}M$ is invertible.
2. The rank of ${}M$ is ${}n$.
3. The rows of ${}M$ are linearly independent.
4. The columns of ${}M$ are linearly independent.

Proof

This follows from fact and from fact.

$\Box$