Rank (linear algebra)
Rank measures the dimension of a matrix's column or row space.
Rank is a fundamental concept in linear algebra, defined as the dimension of the vector space spanned by the columns (or rows) of a matrix. It measures the nondegenerateness of the system of linear equations and linear transformations encoded by the matrix, and is one of its most fundamental characteristics. The rank corresponds to the maximal number of linearly independent columns, which is identical to the maximal number of linearly independent rows. This number is commonly denoted by notations such as rank(A) or rk(A), and occasionally by the German-derived symbol rg(A). More generally, for a linear map between two vector spaces, the rank is defined as the dimension of its image.
A matrix is said to have full rank when its rank equals the largest possible for its dimensions, which is the lesser of its number of rows and columns. If it does not achieve this, it is termed rank-deficient, with the rank deficiency being the difference between that lesser dimension and the actual rank. A standard method for computing the rank is to reduce the matrix to row echelon form using elementary row operations. These operations preserve the row space and map the column space to an isomorphic space, so the rank of the resulting matrix—equal to the number of pivots or non-zero rows—is the same as the original. In numerical computing, methods like Gaussian elimination can be unreliable for floating-point data; more robust approaches include singular value decomposition (SVD) or rank-revealing QR factorization, though these require a criterion for treating small values as zero. The equality of column rank and row rank is a fundamental result, proven by showing that neither rank changes under elementary row operations, and that further column operations can transform the matrix into an identity matrix bordered by zeros, whose rank is simply its number of non-zero entries.
- field
- Linear algebra
- known_for
- Definition of rank of a matrix and linear map; column rank equals row rank; rank as dimension of image of a linear map
Lore & Background
In linear algebra, the rank of a matrix A is the dimension of the vector space generated by its columns, corresponding to the maximal number of linearly independent columns. This number is also identical to the dimension of the vector space spanned by its rows, a fundamental result known as the equality of column rank and row rank. The rank is commonly denoted by rank(A), rk(A), or rg(A) (from German Rang). More generally, the rank of a linear map between two vector spaces is defined as the dimension of its image. A matrix is said to have full rank if its rank equals the lesser of the number of rows and columns; otherwise it is rank-deficient. The rank deficiency is the difference between that lesser number and the rank. Examples illustrate the concept: a 3x3 matrix with columns [1,0,0], [0,1,1], [1,1,1] has rank 2, as the third column is a linear combination of the first two. A 2x4 matrix with rows [1,1,0,2] and [-1,-1,0,-2] has rank 1, as all columns are linearly dependent. The rank of a matrix equals the rank of its transpose.
Reader's Guide
Rank is a central concept in linear algebra, serving as a measure of the nondegenerateness of a matrix or linear transformation. It determines the dimension of the image of a linear map and the number of linearly independent rows or columns. The equality of column rank and row rank is a fundamental theorem, with multiple proofs. Rank is used to characterize systems of linear equations: a system has a solution if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. It also determines the dimension of the null space via the rank–nullity theorem. Computing rank via row echelon form is a standard method, as elementary row operations preserve row space and map column space isomorphically, so the number of pivots equals the rank. Full rank matrices are invertible (if square) or have maximal possible rank for their dimensions. Rank deficiency indicates linear dependence among rows or columns. The concept extends to linear maps between vector spaces, where rank is the dimension of the image. Overall, rank is a fundamental invariant that appears throughout linear algebra and its applications.
Did You Know?
- The rank of a matrix is the dimension of the vector space spanned by its columns.
- Column rank and row rank are always equal.
- A matrix has full rank if its rank equals the lesser of the number of rows and columns.
- The rank of a linear map is the dimension of its image.
Frequently Asked Questions
Who is Rank (linear algebra)?
Rank is a foundational concept in linear algebra that assigns a single number to a matrix, representing the dimension of the space spanned by its columns or rows. It is one of the most basic invariants used to characterize any linear transformation.
What are Rank (linear algebra)'s powers/role?
Its core ability is measuring how 'full' a linear map is by counting the number of independent directions it actually reaches. A signature trait is that column rank and row rank are always equal, so no matter which side you inspect, you get the same number.
How does Rank (linear algebra)'s story end?
In any concrete problem, Rank resolves to a single non-negative integer no larger than the shorter dimension of the matrix. That final value dictates whether a system of equations is consistent, has a unique solution, or admits infinitely many.
Why is Rank (linear algebra) important?
It is the quickest diagnostic for telling whether a transformation collapses space or preserves dimensionality. Nearly every deeper result in linear algebra—solvability criteria, invertibility tests, the rank-nullity theorem—builds on this one number.
What is Rank (linear algebra)'s closest ally?
The dimension of the image (range) of a linear map is essentially its twin, since rank is defined as exactly that dimension. Together they anchor the rank-nullity theorem, which pairs rank with the nullity of the kernel to account for the full domain.
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