Rank

The rank of a matrix is the number of pivots and the number of dimensions that the column space of a matrix spans.

Description

A matrix A with n columns (or a linear system with n variables) exists in Rⁿ space. However, the column space of A (notated as C(A)) does not necessarily span all of those dimensions. Consider:

julia> using RowEchelon

julia> rref([1 2 3
             1 3 4
             1 4 5])
3×3 Matrix{Float64}:
 1.0  0.0  1.0
 0.0  1.0  1.0
 0.0  0.0  0.0

The elimination of this matrix reveals that there are two pivot columns and a free variable.

The rank of a matrix is the number of pivots. Equivalent, it is the number of dimensions that the column space of a matrix spans. For a transformation represented by a matrix, rank is also the number of dimensions in the transformation's range.

Relation to Bases

A basis vector must be independent. It follows that a matrix of n columns must be of full rank to be a basis for Rⁿ space.

Note that in the above example, A has two independent columns. If the third column is excluded, then clearly the matrix exists in R² space rather than R³ space. But the matrix is now of full rank, and therefore forms a basis for R² space.

Solutions

The rank of a matrix reveals the number of solutions that exist.

If a matrix with n columns has n pivots, it is said to be full column rank. The only null space is the zero vector (i.e. [0 ...]). Ax = b either has one solution, or b is not in the column space of A (i.e., no solution exists). Incidentally, the reduced row echelon form (R) looks like the identity matrix (I) with zero rows interspersed.

If a matrix with m rows has m pivots, it is said to be full row rank. This means that Ax = b can be solved for any b.

If a square matrix has full rank, it meets both of those criteria and inherits all of those properties:

• The only null space is the zero vector.

• Ax = b can be solved for any b.

• Ax = b either has one solution or is not solvable, except property #2 revealed that Ax = b can be solved for any b, so the only logical conclusion is that there is exactly one solution for any given b.

• The reduced row echelon form (R) looks like the identity matrix (I) with zero rows interspersed, except there are no dependent rows, so R is exactly the same as I.

In any other case, Ax = b either can be solved for any b, or b is not in the column space of A (i.e., no solution exists).

Lastly, rank provides a simple method to determine if b is in the column space of A (i.e., a solution exists): compose the augmented matrix [A | b]. If rank(A) = rank([A | b]), then b must be in the column space.

CategoryRicottone