Linear independence, span, rank

Spaces associated with a matrix

Definition

The column space of a matrix is defined as the vector space spanned by columns of the matrix.

Let \(A\) be an \(m \times n\) matrix with

\[A = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix}\]

Then the column space is given by

\[\ColSpace(A) = \{ x \in \FF^m : x = \sum_{i=1}^n \alpha_i a_i \; \text{for some } \alpha_i \in \FF \}.\]

Definition

The row space of a matrix is defined as the vector space spanned by rows of the matrix.

Let \(A\) be an \(m \times n\) matrix with

\[\begin{split}A = \begin{bmatrix} a_1^T \\ a_2^T \\ \vdots \\ a_m^T \end{bmatrix}\end{split}\]

Then the row space is given by

\[\RowSpace(A) = \{ x \in \FF^n : x = \sum_{i=1}^m \alpha_i a_i \; \text{for some } \alpha_i \in \FF \}.\]

Rank

Definition

The column rank of a matrix is defined as the maximum number of columns which are linearly independent. In other words column rank is the dimension of the column space of a matrix.

Definition

The row rank of a matrix is defined as the maximum number of rows which are linearly independent. In other words row rank is the dimension of the row space of a matrix.

Theorem

The column rank and row rank of a matrix are equal.

Definition

The rank of a matrix is defined to be equal to its column rank which is equal to its row rank.

Lemma

For an \(m \times n\) matrix \(A\)

\[0 \leq \Rank(A) \leq \min(m, n).\]

Lemma

The rank of a matrix is 0 if and only if it is a zero matrix.

Definition

An \(m \times n\) matrix \(A\) is called full rank if

\[\Rank (A) = \min(m, n).\]

In other words it is either a full column rank matrix or a full row rank matrix or both.

Lemma

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix then

\[\Rank(AB) \leq \min (\Rank(A), \Rank(B)).\]

Lemma

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix. If \(B\) is of rank \(n\) then

\[\Rank(AB) = \Rank(A).\]

Lemma

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix. If \(A\) is of rank \(n\) then

\[\Rank(AB) = \Rank(B).\]

Lemma

The rank of a diagonal matrix is equal to the number of non-zero elements on its main diagonal.

Proof

The columns which correspond to diagonal entries which are zero are zero columns. Other columns are linearly independent. The number of linearly independent rows is also the same. Hence their count gives us the rank of the matrix.