Definition: Rank. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
- Alternatively, it can be thought of as the dimension of the linear span of the vectors in a matrix
- For a matrix to be full rank, the rank will be equal to the smaller of its dimensions (either rows or columns)
- There are multiple ways to show that some matrix
is or is not full rank - If a matrix can be reduced to have a row with all zeros, this means that matrix does not have full rank
- If the determinant of
, , it is full rank. If not, and , the the matrix is not full rank
- If the rank of a matrix is equal to the number of unknowns in a system of linear equations, then the system has a unique solution
- If the rank is less than the number of unknowns, there are infinitely many solutions