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

$\backslash mathbf\; x\; =\; \backslash bigx\_1,\; x\_2,\; \backslash dots,\; x\_n\; \backslash big.$

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

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

$\backslash mathbf\; x\; =\; \backslash begin\{bmatrix\}\; x\_1,\; x\_2,\; \backslash dots,\; x\_m\; \backslash end\{bmatrix\}^\{\backslash rm\; T\}$

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 $x$ is a row vector, then $x$ and $x^\{\backslash rm\; T\}$ might be denoted as follows.

$\backslash mathbf\; x\; =\; \backslash begin\{bmatrix\}\; x\_1\; \backslash ;\; x\_2\; \backslash ;\; \backslash dots\; \backslash ;\; x\_m\; \backslash end\{bmatrix\}\; \backslash qquad$

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

Operations

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• 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.

Linear algebra | Matrices

Rows | 行向量