Row and column vectors

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In linear algebra, a column vector with ⁠${\displaystyle m}$⁠ elements is an ${\displaystyle m\times 1}$ matrix^[1] consisting of a single column of ⁠${\displaystyle m}$⁠ entries, for example, $${\displaystyle {\boldsymbol {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}.}$$

Similarly, a row vector is a ${\displaystyle 1\times n}$ matrix for some ⁠${\displaystyle n}$⁠, consisting of a single row of ⁠${\displaystyle n}$⁠ entries, $${\displaystyle {\boldsymbol {a}}={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\end{bmatrix}}.}$$ (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: $${\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}}$$ and $${\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}.}$$

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.