Rank - Yousef's Notes

Number of dimensions in the output of a transformation.

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. Given a matrix $A$, the rank of $A$ is denoted as $\rank(A)$ or $\rho(A)$.

Full rank: If the rank of $A$ is equal to the number of rows or columns, the matrix is said to have full rank.
Row rank: The row rank of $A$ is the maximum number of linearly independent rows in $A$
Column rank: The column rank of $A$ is the maximum number of linearly independent columns in $A$. The row rank and column rank of a matrix are always equal, so we can simply refer to the rank of the matrix. For example, consider a $2 \times 2$ matrix:

$$ A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} $$ The rank of $A$ is 1, because the second row is a multiple of the first row, and therefore the rows are not linearly independent. In contrast, consider a $2 \times 2$ matrix: $$ B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$

The rank of $B$ is 2, because the rows are linearly independent, and the matrix has full rank. The rank of a matrix has important implications for:

Linear independence: A set of vectors is linearly independent if and only if the matrix formed by the vectors has full rank.
Solving systems of linear equations: The rank of the coefficient matrix determines the number of free variables in the solution.
Matrix inversion: A matrix is invertible if and only if it has full rank.