Row and column vectors

(Redirected from Row vector)

In linear algebra, a column vector is a column of entries, for example,

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

Similarly, a row vector is a row of entries[1]

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

Throughout, boldface is used for both row and column vectors. The transpose (indicated by T) of a row vector is the column 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 the transpose of a column vector is the row vector

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

The set of all row vectors with n entries 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.

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

or

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

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)
${\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}{\text{ or }}{\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}}$
Alternative notation 1
(commas, transpose signs)
${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}}$
Alternative notation 2
(commas and semicolons, no transpose signs)
${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{m}\end{bmatrix}}}$

Operations

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

The dot product of two column vectors a and b is equivalent to the matrix product of the transpose of a with b,

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

By the symmetry of the dot product, the dot product of two column vectors a and b is also equivalent to the matrix product of the transpose of b with a,

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

The matrix product of a column and a row vector gives the outer product of two vectors a and b, an example of the more general tensor 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,

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

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

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

Matrix transformations

An n × n matrix M can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector v, the product vM is another row vector p:

${\displaystyle vM=p\,.}$

Another n × n matrix Q can act on p,

${\displaystyle pQ=t\,.}$

Then one can write t = p Q = v MQ, so the matrix product transformation 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 × n matrix action, the operation occurs to the left,

${\displaystyle p^{\mathrm {T} }=Mv^{\mathrm {T} }\,,\quad t^{\mathrm {T} }=Qp^{\mathrm {T} }}$,

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