Basics

Rank

The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors).

It also can be shown that the columns (rows) of a square matrix are linearly independent only if the matrix is nonsingular. In other words, the rank of any nonsingular matrix of order n is n.

Nonsingular Matrix

An $n \times n$ matrix $A$ is called nonsingular or invertible matrix if there exists an $n \times n$ matrix $B$ such that

$$AB=BA=I.$$

If $A$ does not have an inverse, $A$ is called singular matrix.

Orthogonal Matrices

Definition. A matrix $A$ is orthogonal if $A^TA=AA^T=I$.

Range

The range (also called the column space or image) of a $m \times n$ matrix $A$ is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.