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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.
:
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.

Contents

Notation

Operations

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.
:
m 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 $x^\{$
m T} might be denoted as follows.
:
x^{
m T} = egin{bmatrix} x_1, x_2, dots, x_m end{bmatrix}

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.