Row vector

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In linear algebra, a row vector is a 1 × n matrix, that is, a matrix consisting of a single row:

<math>\mathbf x = \big[ x_1, x_2, \dots, x_n \big].</math>

The transpose of a row vector is a column vector.

The set of all row vectors forms a vector space which is the dual space to the set of all column vectors.

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.

<math>\mathbf x = \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}^{\rm T}</math>

For further simplification, writers also use the convention of writing both column vectors and row vectors as rows but separating row vector elements with spaces and column vector elements with commas. For example, if <math>x</math> is a row vector, then <math>x</math> and <math>x^{\rm T}</math> might be denoted as follows.

<math>\mathbf x = \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix} \qquad

\mathbf x^{\rm T} = \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}</math>

Operations

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

• The dot product in a Euclidean space involves both taking the transpose of a column vector and multiplying the resulting row vector with another column vector.fr:Rows