Row vector
Encyclopedia
In linear algebra
, a row vector or row matrix is a 1 × n matrix
, that is, a matrix consisting of a single row:
The transpose
of a row vector is a column vector:
The set of all row vectors forms a vector space
which acts like the dual space
to the set of all column vectors, in the sense that any linear functional on the space of column vectors (i.e. any element of the dual space) can be represented uniquely as a dot product with a specific row vector.
Then p is also a row vector and may present to another n by n matrix Q:
Conveniently, one can write t = p Q = v MQ telling us that the matrix product transformation MQ can take v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs.
In contrast, when a column vector is transformed to become another column under an n by n matrix action, the operation occurs to the left:
leading to the algebraic expression QM v for the composed output from v input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation. The natural bias to read left-to-right, as subsequent transformations are applied in linear algebra, stands against column vector inputs.
Nevertheless, using the transpose
operation these differences between inputs of a row or column nature are resolved by an antihomomorphism
between the groups arising on the two sides. The technical construction uses the dual space
associated with a vector space to develop the transpose of a linear map.
For an instance where this row vector input convention has been used to good effect see Raiz Usmani (1987), where on page 106 the convention allows the statement "The product mapping ST of U into W [is given] by:
."
(The Greek letters represent row vectors).
Ludwik Silberstein
used row vectors for spacetime events; he applied Lorentz transformation matrices on the right in his Theory of Relativity in 1914 (see page 143).
In 1963 when McGraw-Hill
published Differential Geometry by Heinrich Guggenheimer
of the University of Minnesota
, he uses the row vector convention in chapter 5, "Introduction to transformation groups" (eqs. 7a,9b and 12 to 15).
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a row vector or row matrix is a 1 × n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, that is, a matrix consisting of a single row:
The transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
of a row vector is a column vector:
The set of all row vectors forms a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
which acts like the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
to the set of all column vectors, in the sense that any linear functional on the space of column vectors (i.e. any element of the dual space) can be represented uniquely as a dot product with a specific row vector.
Notation
Row vectors are sometimes written using the following non-standard notation:Operations
- Matrix multiplicationMatrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
involves the action of multiplying each row vector of one matrix by each column vector of another matrix.
- The dot productDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of two vectors a and b is equivalent to multiplying the row vector representation of a by the column vector representation of b:
Preferred input vectors for matrix transformations
Frequently a row vector presents itself for an operation within n-space expressed by an n by n matrix M:- v M = p.
Then p is also a row vector and may present to another n by n matrix Q:
- p Q = t.
Conveniently, one can write t = p Q = v MQ telling us that the matrix product transformation MQ can take v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs.
In contrast, when a column vector is transformed to become another column under an n by n matrix action, the operation occurs to the left:
- p = M v and t = Q p ,
leading to the algebraic expression QM v for the composed output from v input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation. The natural bias to read left-to-right, as subsequent transformations are applied in linear algebra, stands against column vector inputs.
Nevertheless, using the transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
operation these differences between inputs of a row or column nature are resolved by an antihomomorphism
Antihomomorphism
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection from an object to...
between the groups arising on the two sides. The technical construction uses the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
associated with a vector space to develop the transpose of a linear map.
For an instance where this row vector input convention has been used to good effect see Raiz Usmani (1987), where on page 106 the convention allows the statement "The product mapping ST of U into W [is given] by:
."(The Greek letters represent row vectors).
Ludwik Silberstein
Ludwik Silberstein
Ludwik Silberstein was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework...
used row vectors for spacetime events; he applied Lorentz transformation matrices on the right in his Theory of Relativity in 1914 (see page 143).
In 1963 when McGraw-Hill
McGraw-Hill
The McGraw-Hill Companies, Inc., is a publicly traded corporation headquartered in Rockefeller Center in New York City. Its primary areas of business are financial, education, publishing, broadcasting, and business services...
published Differential Geometry by Heinrich Guggenheimer
Heinrich Guggenheimer
Heinrich Walter Guggenheimer is an American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity. He has also contributed volumes on Jewish sacred literature....
of the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
, he uses the row vector convention in chapter 5, "Introduction to transformation groups" (eqs. 7a,9b and 12 to 15).