= Rank =

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

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== Rank and Dimension ==

A matrix with ''n'' '''dimensions''' exists in ''R^n^'' space. However, the '''column space''' of that same matrix does not necessarily exist in the same number of dimensions.

Consider a matrix ''A'' like:

{{{
┌      ┐
│ 1 2 3│
│ 1 3 4│
│ 1 4 5│
└      ┘
}}}

The third column vector can be trivially shown to not be '''independent'''; it is a sum of the first and second column vectors. Correspondingly, the [[LinearAlgebra/Elimination|eliminated]] form of the matrix has two pivots and a free variable. As a direct consequence of this being a square matrix (''n x n''), this also means that there are two pivot rows and a zero row.

{{{
┌         ┐
│ [1] 2  3│
│  0 [1] 1│
│  0  0  0│
└         ┘
}}}

''A'' is not a [[LinearAlgebra/Basis|basis]] for ''R^3^'' space. The requirements for a basis are that each column be independent and that the matrix span all dimensions.

However, this matrix ''does'' contain two independent columns. If those are split off like:

{{{
┌    ┐
│ 1 2│
│ 1 3│
│ 1 4│
└    ┘
}}}

Then this new matrix is a basis for the column space of ''A'' (sometimes notated as ''C(A)''), which happens to be 2 dimensional.

Effectively, this matrix expresses a 2-dimensional plane that exists within a 3-dimensional space, but it is bound to that plane and cannot vary across the third dimension.

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== Categories for Solutions ==

If a matrix has a pivot in each column, it is said to be '''full column rank'''. The only [[LinearAlgebra/NullSpace|null space]] is the zero vector (i.e. ''[0 ...]''). ''Ax = b'' either has one solution or is not solvable. Incidentally, the [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reduced row echelon form]] (''R'') looks like the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] (''I'') with some number of zero rows.

If a matrix has a pivot in each row, it is said to be '''full row rank'''. This only means that ''Ax = b'' can be solved for any ''b''.

If a square matrix has '''full rank''', the reduced row echelon form (''R'') of the matrix is the identity matrix (''I''). The only null space is the zero vector; ''Ax = b'' can be solved for any ''b''; there is exactly one solution for any given ''b''.

If a matrix is none of the above, ''Ax = b'' either can be solved for any ''b'' or is not solvable.



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