Matrix rank
The rank of a matrix is the dimension of its column space, equal to the dimension of its row space.
Definition
The rank of a matrix is the dimension of its column space, equal to the dimension of its row space.
Intuition
Rank counts how many independent output directions the matrix can produce.
Worked example
The matrix with rows (1, 2) and (2, 4) has rank 1 because one row is twice the other.
The math
For an matrix, .
In machine learning
Rank helps detect redundant measurements, singular systems, and the dimension of a low-rank approximation.
Go deeper
Sources
More in Linear algebra
Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.