Row and column vectors

Inlinear algebra,acolumn vectorwith⁠ $m$ ⁠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, arow vectoris 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.)

Thetranspose(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 givenfield(such as thereal numbers) forms an $n$ -dimensionalvector 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 thedual spaceof 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 withcommasand 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)	${\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}$	${\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}{\text{ or }}{\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}$
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)	${\begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}}$	${\begin{bmatrix}x_{1};x_{2};\dots;x_{m}\end{bmatrix}}$

Operations

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

Thedot productof two column vectors $a, b$ ,considered as elements of a coordinate space, is equal to the matrix product of the transpose of $a$ with $b$ ,

$\mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\intercal }\mathbf {b} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}b_{1}\\\vdots \\b_{n}\end{bmatrix}}=a_{1}b_{1}+\cdots +a_{n}b_{n}\,,$

By the symmetry of the dot product, thedot productof two column vectors $a, b$ is also equal to the matrix product of the transpose of $b$ with $a$ ,

$\mathbf {b} \cdot \mathbf {a} =\mathbf {b} ^{\intercal }\mathbf {a} ={\begin{bmatrix}b_{1}&\cdots &b_{n}\end{bmatrix}}{\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}=a_{1}b_{1}+\cdots +a_{n}b_{n}\,.$

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

$\mathbf {a} \otimes \mathbf {b} =\mathbf {a} \mathbf {b} ^{\intercal }={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}{\begin{bmatrix}b_{1}&b_{2}&b_{3}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\\\end{bmatrix}}\,,$

which is thetransposeof the matrix product of the column vector representation of $b$ and the row vector representation of $a$ ,

$\mathbf {b} \otimes \mathbf {a} =\mathbf {b} \mathbf {a} ^{\intercal }={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}{\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}}={\begin{bmatrix}b_{1}a_{1}&b_{1}a_{2}&b_{1}a_{3}\\b_{2}a_{1}&b_{2}a_{2}&b_{2}a_{3}\\b_{3}a_{1}&b_{3}a_{2}&b_{3}a_{3}\\\end{bmatrix}}\,.$

Matrix transformations

An $n \times n$ matrix $M$ can represent alinear mapand act on row and column vectors as the linear map'stransformation matrix.For a row vector $v$ ,the product $v M$ is another row vector $p$ :

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

Another $n \times n$ matrix $Q$ can act on $p$ ,

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

Then one can write $t = p Q = v MQ$ ,so thematrix producttransformation $MQ$ maps $v$ directly to $t$ .Continuing with row vectors, matrix transformations further reconfiguring $n$ -space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under an $n \times n$ matrix action, the operation occurs to the left,

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

leading to the algebraic expression $QM v T$ for the composed output from $v T$ input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.

Notes

^Artin, Michael (1991).Algebra.Englewood Cliffs, NJ: Prentice-Hall. p. 2.ISBN 0-13-004763-5.

References

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 originalon 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

[Artin-1] Artin, Michael (1991).Algebra.Englewood Cliffs, NJ: Prentice-Hall. p. 2.ISBN 0-13-004763-5.

[1]

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Notes

References