In
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.
...
, a column vector with
elements is an
matrix
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 ...
consisting of a single column of
entries, for example,
Similarly, a row vector is a
matrix for some
, consisting of a single row of
entries,
(Throughout this article, boldface is used for both row and column vectors.)
The
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
(indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector:
and
The set of all row vectors with ''n'' entries in a given
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
(such as the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
) forms an ''n''-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
; similarly, the set of all column vectors with ''m'' entries forms an ''m''-dimensional vector space.
The space of row vectors with ''n'' entries can be regarded as the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of the space of column vectors with ''n'' entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.
Notation
To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.
:
or
:
Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
s and column vector elements with
semicolon
The semicolon or semi-colon is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought. When a ...
s (see alternative notation 2 in the table below).
Operations
Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
involves the action of multiplying each row vector of one matrix by each column vector of another matrix.
The
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of two column vectors a and b, considered as elements of a coordinate space, is equal to the matrix product of the transpose of a with b,
:
By the symmetry of the dot product, the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of two column vectors a and b is also equal to the matrix product of the transpose of b with a,
:
The matrix product of a column and a row vector gives the
outer product of two vectors a and b, an example of the more general
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
. The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product,
:
which is the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of the matrix product of the column vector representation of b and the row vector representation of a,
:
Matrix transformations
An ''n'' × ''n'' matrix ''M'' can represent a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
and act on row and column vectors as the linear map's
transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
. For a row vector ''v'', the product ''vM'' is another row vector ''p'':
:
Another ''n'' × ''n'' matrix ''Q'' can act on ''p'',
:
Then one can write ''t'' = ''p Q'' = ''v MQ'', so the
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
transformation ''MQ'' maps ''v'' directly to ''t''. Continuing with row vectors, matrix transformations further reconfiguring ''n''-space can be applied to the right of previous outputs.
When a column vector is transformed to another column vector under an ''n'' × ''n'' matrix action, the operation occurs to the left,
:
,
leading to the algebraic expression ''QM v''
T for the composed output from ''v''
T input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.
See also
*
Covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...
*
Index notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to th ...
*
Vector of ones
In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below:
:J_2 = \begin
1 & 1 \\
1 & 1
\end;\quad
J_3 = \begin
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end;\qu ...
*
Single-entry vector
In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as E_. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is
E_ = \begi ...
*
Standard unit vector
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
*
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
Notes
References
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{{Linear algebra
Linear algebra
Matrices
Vectors (mathematics and physics)