linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

, two

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two ''m'' × ''n'' matrices are row equivalent if and only if they have the same

row space Row or ROW may refer to: Exercise *Rowing, or a form of aquatic movement using oars *Row (weight-lifting), a form of weight-lifting exercise Math *Row vector, a 1 × ''n'' matrix in linear algebra. *Row (database), a single, implicitly structured ...

. The concept is most commonly applied to matrices that represent

systems of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...

, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same

null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...

. Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a

tilde The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) i ...

(~). There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...

Elementary row operations

elementary row operation In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multipl ...

is any one of the following moves: # Swap: Swap two rows of a matrix. # Scale: Multiply a row of a matrix by a nonzero constant. # Pivot: Add a multiple of one row of a matrix to another row. Two matrices ''A'' and ''B'' are row equivalent if it is possible to transform ''A'' into ''B'' by a sequence of elementary row operations.

Row space

The row space of a matrix is the set of all possible linear combinations of its row vectors. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Two ''m'' × ''n'' matrices are row equivalent if and only if they have the same row space. For example, the matrices :

\begin1 & 0 & 0 \\ 0 & 1 & 1\end
\;\;\;\;\text\;\;\;\;
\begin1 & 0 & 0 \\ 1 & 1 & 1\end

are row equivalent, the row space being all vectors of the form

\begina & b & b\end

. The corresponding systems of homogeneous equations convey the same information: :

\beginx = 0 \\ y+z=0\end\;\;\;\;\text\;\;\;\;\begin x=0 \\ x+y+z=0.\end

In particular, both of these systems imply every equation of the form

ax+by+bz=0.\,

Equivalence of the definitions

The fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations: # Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space. # Any matrix can be reduced by elementary row operations to a matrix in

reduced row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian e ...

. # Two matrices in reduced row echelon form have the same row space if and only if they are equal. This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form.

Additional properties

* Because the

of a matrix is the

orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

of the

, two matrices are row equivalent if and only if they have the same null space. * The

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

of a matrix is equal to the

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. * A matrix is

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

if and only if it is row equivalent to the identity matrix. * Matrices ''A'' and ''B'' are row equivalent if and only if there exists an invertible matrix ''P'' such that ''A=PB''.

References

* * * * * * *

External links