# Full-rank

A matrix is **full-rank** when all of its rows are linearly independent. Otherwise, it is **rank-deficient**.

For example, the mapping matrix [⟨41 65 95 115] ⟨31 49 72 87] ⟨19 30 44 53]⟩ is full-rank. We can check this by putting it into Hermite normal form (HNF), [⟨1 0 0 -5] ⟨0 1 0 2] ⟨0 0 1 2]⟩, and observing that there are no rows of all zeros at the bottom (this is the mapping for marvel temperament). On the other hand, [⟨41 65 95 115] ⟨31 49 72 87] ⟨10 16 23 28]⟩ is rank-deficient, because its HNF is [⟨1 1 3 3] ⟨0 6 -7 -2] ⟨0 0 0 0]⟩, so we can see a row of all zeros has been produced at the bottom.

In Wolfram Language, an even quicker check for full-rank is possible, using `MatrixRank[]`

, which will give you the count of linearly independent rows of a matrix. If this is less than the count of rows, the matrix is rank-deficient.

You can guarantee a full-rank result by putting a matrix into canonical form.

One could generalize this notion to full-nullity and nullity-deficient when speaking of the linear independence of columns of a comma basis. And therefore further generalize the notion to grade, with full-grade and grade-deficient.