Editing Row and column vectors (section)

{{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.