Rank of a matrix

Dependencies:

  1. Field
  2. RREF is unique
  3. Row space
  4. Basis of a vector space
  5. Row equivalent matrices have the same row space

The rank of a matrix over a field is the number of non-zero rows in the RREF of the matrix.

The rank of a set $X$ of vectors is the rank of the matrix whose rows are $X$.

Since any 2 row-equivalent matrices have the same RREF, they also have the same rank.

A matrix is called full-rank when its rank equals the number of rows in it.

Equivalently, the rank of a matrix is the dimension of its row space.

Proof of equivalence of definitions

Let $A$ be a matrix. Let $R$ be its RREF. Since the RREF is obtained by elementary row operations on $A$, $A$ and $R$ are row-equivalent. Row-equivalent matrices have the same row space, so $A$ and $R$ have the same row space, and consequently have the same dimension for the row space.

The non-zero rows in $R$ are linearly independent, and they span the row space of $R$. Hence, those rows form a basis of $R$'s row space. Hence, the dimension of $R$'s row space equals the number of non-zero rows in $R$.

Dependency for:

  1. A matrix is full-rank iff its rows are linearly independent
  2. Full-rank square matrix is invertible
  3. A matrix is full-rank iff its determinant is non-0
  4. Rank of a homogenous system of linear equations
  5. Condition for polyhedral cone to be pointed
  6. Representing point in full-rank polyhedron
  7. Bounded section of pointed cone
  8. BFS is vertex
  9. Representing point in pointed polyhedral cone
  10. Condition for existence of BFS in a polyhedron
  11. LP is optimized at BFS
  12. Extreme point iff BFS
  13. Pointing a polyhedron
  14. Point in polytope is convex combination of BFS

Info:

Transitive dependencies:

  1. /linear-algebra/matrices/gauss-jordan-algo
  2. /linear-algebra/vector-spaces/condition-for-subspace
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Field
  8. Vector Space
  9. Linear independence
  10. Span
  11. Integral Domain
  12. Comparing coefficients of a polynomial with disjoint variables
  13. Semiring
  14. Matrix
  15. Stacking
  16. System of linear equations
  17. Product of stacked matrices
  18. Matrix multiplication is associative
  19. Reduced Row Echelon Form (RREF)
  20. Elementary row operation
  21. Every elementary row operation has a unique inverse
  22. Row equivalence of matrices
  23. Matrices over a field form a vector space
  24. Row space
  25. Row equivalent matrices have the same row space
  26. RREF is unique
  27. Identity matrix
  28. Inverse of a matrix
  29. Inverse of product
  30. Elementary row operation is matrix pre-multiplication
  31. Row equivalence matrix
  32. Equations with row equivalent matrices have the same solution set
  33. Basis of a vector space
  34. Linearly independent set is not bigger than a span
  35. Homogeneous linear equations with more variables than equations
  36. Rank of a homogenous system of linear equations