Row and column vectors

In linear algebra, a column vector with $m$ elements is an $m\times 1$ matrix 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.

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