Editing Row and column vectors

{{short description|Matrix consisting of a single row or column}}
{{more footnotes|date=November 2022}}

In [[linear algebra]], a '''column vector''' with {{tmath|m}} elements is an <math>m \times 1</math> [[Matrix_(mathematics)|matrix]]<ref name="Artin">{{cite book |last1=Artin |first1=Michael |title=Algebra |date=1991 |publisher=Prentice-Hall |location=Englewood Cliffs, NJ |isbn=0-13-004763-5 |page=2}}</ref> consisting of a single column of {{tmath|m}} entries, for example,
<math display="block">\boldsymbol{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}.</math>

Similarly, a '''row vector''' is a <math>1 \times n</math> matrix for some {{tmath|n}}, consisting of a single row of {{tmath|n}} entries,
<math display="block">\boldsymbol a = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix}. </math>
(Throughout this article, boldface is used for both row and column vectors.)
 
The [[transpose]] (indicated by {{math|T}}) of any row vector is a column vector, and the transpose of any column vector is a row vector:
<math display="block">\begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}</math>
and
<math display="block">\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}^{\rm T} = \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}.</math>

The set of all row vectors with {{mvar|n}} entries in a given [[field (mathematics)|field]] (such as the [[real numbers]]) forms an {{mvar|n}}-dimensional [[vector space]]; similarly, the set of all column vectors with {{mvar|m}} entries forms an {{mvar|m}}-dimensional vector space.

The space of row vectors with {{mvar|n}} entries can be regarded as the [[dual space]] of the space of column vectors with {{mvar|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.

<math display="block">\boldsymbol{x} = \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T}</math>

or

<math display="block">\boldsymbol{x} = \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}^{\rm T}</math>

Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with [[comma]]s and column vector elements with [[semicolon]]s (see alternative notation 2 in the table below).{{fact|date=March 2021}}
{| class="wikitable"
|-
! !! Row vector !! Column vector
|-
| '''Standard matrix notation'''<br/>(array spaces, no commas, transpose signs)
| align=center| <math> \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix} </math> 
| align=center| <math> \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix} \text{ or } \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T} </math> 
|-
| '''Alternative notation 1'''<br/>(commas, transpose signs)
| align=center| <math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix} </math>
| align=center| <math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}^{\rm T} </math>
|-
| '''Alternative notation 2'''<br/>(commas and semicolons, no transpose signs)
| align=center| <math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix} </math>
| align=center| <math> \begin{bmatrix} x_1; x_2; \dots; x_m \end{bmatrix} </math>
|}

== Operations ==

[[Matrix multiplication]] involves the action of multiplying each row vector of one matrix by each column vector of another matrix. 

The [[dot product]] of two column vectors {{math|'''a''', '''b'''}}, considered as elements of a coordinate space, is equal to the matrix product of the transpose of {{math|'''a'''}} with {{math|'''b'''}},

<math display="block">\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^\intercal \mathbf{b} = \begin{bmatrix}
    a_1  & \cdots  & a_n
\end{bmatrix} \begin{bmatrix} 
    b_1 \\ \vdots \\ b_n
\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n \,, </math>

By the symmetry of the dot product, the [[dot product]] of two column vectors {{math|'''a''', '''b'''}} is also equal to the matrix product of the transpose of {{math|'''b'''}} with {{math|'''a'''}},

<math display="block">\mathbf{b} \cdot \mathbf{a} = \mathbf{b}^\intercal \mathbf{a} = \begin{bmatrix}
    b_1  & \cdots  & b_n
\end{bmatrix}\begin{bmatrix} 
    a_1 \\ \vdots \\ a_n
\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n\,. </math>

The matrix product of a column and a row vector gives the [[outer product]] of two vectors {{math|'''a''', '''b'''}}, an example of the more general [[tensor product]]. The matrix product of the column vector representation of {{math|'''a'''}} and the row vector representation of {{math|'''b'''}} gives the components of their dyadic product,

<math display="block">\mathbf{a} \otimes \mathbf{b} = \mathbf{a} \mathbf{b}^\intercal = \begin{bmatrix}
    a_1 \\ a_2 \\ a_3
\end{bmatrix}\begin{bmatrix} 
    b_1 & b_2 & b_3
\end{bmatrix} = \begin{bmatrix} 
a_1 b_1 & a_1 b_2 & a_1 b_3 \\
a_2 b_1 & a_2 b_2 & a_2 b_3 \\
a_3 b_1 & a_3 b_2 & a_3 b_3 \\
\end{bmatrix} \,, </math>

which is the [[transpose]] of the matrix product of the column vector representation of {{math|'''b'''}} and the row vector representation of {{math|'''a'''}},

<math display="block">\mathbf{b} \otimes \mathbf{a} = \mathbf{b} \mathbf{a}^\intercal = \begin{bmatrix}
    b_1 \\ b_2 \\ b_3
\end{bmatrix}\begin{bmatrix} 
    a_1 & a_2 & a_3
\end{bmatrix} = \begin{bmatrix} 
b_1 a_1 & b_1 a_2 & b_1 a_3 \\
b_2 a_1 & b_2 a_2 & b_2 a_3 \\
b_3 a_1 & b_3 a_2 & b_3 a_3 \\
\end{bmatrix} \,. </math>

==Matrix transformations==
{{main|Transformation matrix}}
An {{math|''n'' × ''n''}} matrix {{mvar|M}} can represent a [[linear map]] and act on row and column vectors as the linear map's [[transformation matrix]]. For a row vector {{math|'''v'''}}, the product {{math|'''v'''''M''}} is another row vector {{math|'''p'''}}: 

<math display="block">\mathbf{v} M = \mathbf{p} \,.</math>

Another {{math|''n'' × ''n''}} matrix {{mvar|Q}} can act on {{math|'''p'''}},

<math display="block"> \mathbf{p} Q = \mathbf{t} \,. </math>

Then one can write {{math|1='''t''' = '''p'''''Q'' = '''v'''''MQ''}}, so the [[matrix product]] transformation {{mvar|MQ}} maps {{math|'''v'''}} directly to {{math|'''t'''}}.  Continuing with row vectors, matrix transformations further reconfiguring {{mvar|n}}-space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under an {{math|''n'' × ''n''}} matrix action, the operation occurs to the left,

<math display="block"> \mathbf{p}^\mathrm{T} = M \mathbf{v}^\mathrm{T} \,,\quad \mathbf{t}^\mathrm{T} = Q \mathbf{p}^\mathrm{T},</math>

leading to the algebraic expression {{math|''QM'' '''v'''<sup>T</sup>}}  for the composed output from {{math|'''v'''<sup>T</sup>}} 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]]
* [[Index notation]]
* [[Vector of ones]]
* [[Single-entry vector]]
* [[Standard unit vector]]
* [[Unit vector]]

== Notes ==

{{reflist}}

== References ==

{{see also|Linear algebra#Further reading}}

* {{Citation
 | last = Axler
 | first = Sheldon Jay
 | date = 1997
 | title = Linear Algebra Done Right
 | publisher = Springer-Verlag
 | edition = 2nd
 | isbn = 0-387-98259-0
}}
* {{Citation
 | last = Lay
 | first = David C.
 | date = August 22, 2005
 | title = Linear Algebra and Its Applications
 | publisher = Addison Wesley
 | edition = 3rd
 | isbn = 978-0-321-28713-7
}}
* {{Citation
 |last        = Meyer
 |first       = Carl D.
 |date        = February 15, 2001
 |title       = Matrix Analysis and Applied Linear Algebra
 |publisher   = Society for Industrial and Applied Mathematics (SIAM)
 |isbn        = 978-0-89871-454-8
 |url         = http://www.matrixanalysis.com/DownloadChapters.html
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20010301161440/http://matrixanalysis.com/DownloadChapters.html
 |archive-date = March 1, 2001
}}
* {{Citation
 | last = Poole
 | first = David
 | date = 2006
 | title = Linear Algebra: A Modern Introduction
 | publisher = Brooks/Cole
 | edition = 2nd
 | isbn = 0-534-99845-3
}}
* {{Citation
 | last = Anton
 | first = Howard
 | date = 2005
 | title = Elementary Linear Algebra (Applications Version)
 | publisher = Wiley International
 | edition = 9th
}}
* {{Citation
 | last = Leon
 | first = Steven J.
 | date = 2006
 | title = Linear Algebra With Applications
 | publisher = Pearson Prentice Hall
 | edition = 7th
}}

{{Linear algebra}}

[[Category:Linear algebra]]
[[Category:Matrices (mathematics)]]
[[Category:Vectors (mathematics and physics)]]