Definitions

# Column vector

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

$mathbf\left\{x\right\} = begin\left\{bmatrix\right\} x_1 x_2 vdots x_m end\left\{bmatrix\right\}$

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.

$mathbf\left\{x\right\} = begin\left\{bmatrix\right\} x_1 ; x_2 ; dots ; x_m end\left\{bmatrix\right\}^\left\{rm T\right\}$
or
$mathbf\left\{x\right\} = begin\left\{bmatrix\right\} x_1, x_2, dots, x_m end\left\{bmatrix\right\}^\left\{rm T\right\}$

For further simplification, 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). This alternative notation is used, for instance, in MATLAB, a widely used programming language specifically designed to simplify computations involving matrices.

Row vector Column vector
Standard matrix notation $begin\left\{bmatrix\right\} x_1 ; x_2 ; dots ; x_m end\left\{bmatrix\right\}$ $begin\left\{bmatrix\right\} x_1 x_2 vdots x_m end\left\{bmatrix\right\} text\left\{ or \right\} begin\left\{bmatrix\right\} x_1 ; x_2 ; dots ; x_m end\left\{bmatrix\right\}^\left\{rm T\right\}$
Alternative notation 1 $begin\left\{bmatrix\right\} x_1, x_2, dots, x_m end\left\{bmatrix\right\} qquad$ $begin\left\{bmatrix\right\} x_1, x_2, dots, x_m end\left\{bmatrix\right\}^\left\{rm T\right\}$
Alternative notation 2 $begin\left\{bmatrix\right\} x_1, x_2, dots, x_m end\left\{bmatrix\right\} qquad$ $begin\left\{bmatrix\right\} x_1; x_2; dots; x_m end\left\{bmatrix\right\}$

## 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 vectors a and b is equivalent to multiplying the row vector representation of a by the column vector representation of b:

$mathbf\left\{a\right\} cdot mathbf\left\{b\right\} = begin\left\{bmatrix\right\}$
`   a_1  & a_2  & a_3`
end{bmatrix}begin{bmatrix}
`   b_1  b_2  b_3`
end{bmatrix}.

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