Column vector

In linear algebra, a column vector with $m$ elements is an $m\times 1$ matrix^[1] consisting of a single column of $m$ entries, for example,

{\boldsymbol {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}.

Similarly, a row vector is a $1\times n$ matrix for some $n$ , consisting of a single row of $n$ entries,

{\boldsymbol {a}}={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\end{bmatrix}}.

(Throughout this article, boldface is used for both row and column vectors.)

The transpose (indicated by $T$ ) of any row vector is a column vector, and the transpose of any column vector is a row vector:

{\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}

and

{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}.

The set of all row vectors with $n$ entries in a given field (such as the real numbers) forms an $n$ -dimensional vector space; 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 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.

{\boldsymbol {x}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}

or

{\boldsymbol {x}}={\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}

Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below).^{[citation needed]}

	Row vector	Column vector
Standard matrix notation (array spaces, no commas, transpose signs)	${\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}$	${\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}{\text{ or }}{\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}$
Alternative notation 1 (commas, transpose signs)	${\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}$	${\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}$
Alternative notation 2 (commas and semicolons, no transpose signs)	${\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}$	${\begin{bmatrix}x_{1};x_{2};\dots ;x_{m}\end{bmatrix}}$

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