Column vector

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In linear algebra, a column vector is an m × 1 matrix, i.e. a matrix consisting of a single column of [m] elements.

[\mathbf x = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end]

The transpose of a column vector is a row vector and vice versa.

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

[\mathbf x = \begin x_1, x_2, \dots, x_m \end^]

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^] might be denoted as follows.

[\mathbf x = \begin x_1 \; x_2 \; \dots \; x_m \end \qquad \mathbf x^ = \begin x_1, x_2, \dots, x_m \end]

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.

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