# Full-rank

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A matrix is **full-rank** when either all of its rows are linearly independent or all of its columns 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.

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

## See also

- Linear dependence#Rank-deficiency and full-rank: for a more in-depth textbook tutorial style look at this concept and how it relates to RTT
- Related terminology on Wikipedia