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Inlinear algebra, acolumn vector with⁠${\displaystyle m}$⁠ elements is an${\displaystyle m\times 1}$matrix^[1] consisting of a single column of⁠${\displaystyle m}$⁠ entries, for example,$${\displaystyle \mathit{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_m\end{array}\right].}$$

Similarly, arow vector is a${\displaystyle 1\times n}$ matrix for some⁠${\displaystyle n}$⁠, consisting of a single row of⁠${\displaystyle n}$⁠ entries,(Throughout this article, boldface is used for both row and column vectors.)

Thetranspose (indicated byT) of any row vector is a column vector, and the transpose of any column vector is a row vector:$${\displaystyle {\left[\begin{array}{c}{x}_1{x}_2\dots x_m\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_m\end{array}\right]}$$and$${\displaystyle {\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_m\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{c}{x}_1{x}_2\dots x_m\end{array}\right].}$$

The set of all row vectors withn entries in a givenfield (such as thereal numbers) forms ann-dimensionalvector space; similarly, the set of all column vectors withm entries forms anm-dimensional vector space.

The space of row vectors withn entries can be regarded as thedual space of the space of column vectors withn entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.

To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

$${\displaystyle \mathit{x}={\left[\begin{array}{c}{x}_1{x}_2\dots x_m\end{array}\right]}^{\mathrm{T}}}$$

or

$${\displaystyle \mathit{x}={\left[\begin{array}{c}{x}_1,x_2,\dots ,x_m\end{array}\right]}^{\mathrm{T}}}$$

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

	Row vector	Column vector
Standard matrix notation (array spaces, no commas, transpose signs)	${\displaystyle \left[\begin{array}{c}{x}_1{x}_2\dots x_m\end{array}\right]}$	${\displaystyle \left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_m\end{array}\right]\text{ or }{\left[\begin{array}{c}{x}_1{x}_2\dots x_m\end{array}\right]}^{\mathrm{T}}}$
Alternative notation 1 (commas, transpose signs)	${\displaystyle \left[\begin{array}{c}{x}_1,x_2,\dots ,x_m\end{array}\right]}$	${\displaystyle {\left[\begin{array}{c}{x}_1,x_2,\dots ,x_m\end{array}\right]}^{\mathrm{T}}}$
Alternative notation 2 (commas and semicolons, no transpose signs)	${\displaystyle \left[\begin{array}{c}{x}_1,x_2,\dots ,x_m\end{array}\right]}$	${\displaystyle \left[\begin{array}{c}{x}_1;x_2;\dots ;x_m\end{array}\right]}$

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

Thedot product of two column vectorsa,b, considered as elements of a coordinate space, is equal to the matrix product of the transpose ofa withb,

By the symmetry of the dot product, thedot product of two column vectorsa,b is also equal to the matrix product of the transpose ofb witha,

The matrix product of a column and a row vector gives theouter product of two vectorsa,b, an example of the more generaltensor product. The matrix product of the column vector representation ofa and the row vector representation ofb gives the components of their dyadic product,

which is thetranspose of the matrix product of the column vector representation ofb and the row vector representation ofa,

Ann ×n matrixM can represent alinear map and act on row and column vectors as the linear map'stransformation matrix. For a row vectorv, the productvM is another row vectorp:

$${\displaystyle \mathbf{v}M=\mathbf{p}.}$$

Anothern ×n matrixQ can act onp,

$${\displaystyle \mathbf{p}Q=\mathbf{t}.}$$

Then one can writet =pQ =vMQ, so thematrix product transformationMQ mapsv directly tot. Continuing with row vectors, matrix transformations further reconfiguringn-space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under ann ×n matrix action, the operation occurs to the left,

$${\displaystyle {\mathbf{p}}^{\mathrm{T}}=M{\mathbf{v}}^{\mathrm{T}},{\mathbf{t}}^{\mathrm{T}}=Q{\mathbf{p}}^{\mathrm{T}},}$$

leading to the algebraic expressionQMv^T for the composed output fromv^T input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.

• Covariance and contravariance of vectors
• Index notation
• Vector of ones
• Single-entry vector
• Standard unit vector
• Unit vector

• Axler, Sheldon Jay (1997),Linear Algebra Done Right (2nd ed.), Springer-Verlag,ISBN 0-387-98259-0
• Lay, David C. (August 22, 2005),Linear Algebra and Its Applications (3rd ed.), Addison Wesley,ISBN 978-0-321-28713-7
• Meyer, Carl D. (February 15, 2001),Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM),ISBN 978-0-89871-454-8, archived fromthe original on March 1, 2001
• Poole, David (2006),Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole,ISBN 0-534-99845-3
• Anton, Howard (2005),Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
• Leon, Steven J. (2006),Linear Algebra With Applications (7th ed.), Pearson Prentice Hall

• Linear algebra
• Matrices
• Vectors (mathematics and physics)

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