Editing Matrix multiplication (section)

====Geometric rotations====
{{See also|Rotation matrix}}
Using a [[Cartesian coordinate]] system in a Euclidean plane, the [[rotation (mathematics)|rotation]] by an angle <math>\alpha</math> around the [[origin (mathematics)|origin]] is a linear map.
More precisely, 
<math display="block"> \begin{bmatrix} x' \\ y' \end{bmatrix} =
 \begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix},</math>
where the source point <math>(x,y)</math> and its image <math>(x',y')</math> are written as column vectors.

The composition of the rotation by <math>\alpha</math> and that by <math>\beta</math> then corresponds to the matrix product
<math display="block">\begin{bmatrix} \cos \beta & - \sin \beta \\ \sin \beta & \cos \beta \end{bmatrix} 
       \begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}
     = \begin{bmatrix} \cos \beta \cos \alpha - \sin \beta \sin \alpha & - \cos \beta \sin \alpha - \sin \beta \cos \alpha \\
                       \sin \beta \cos \alpha + \cos \beta \sin \alpha & - \sin \beta \sin \alpha + \cos \beta \cos \alpha \end{bmatrix}
     = \begin{bmatrix} \cos (\alpha+\beta) & - \sin(\alpha+\beta) \\ \sin(\alpha+\beta) & \cos(\alpha+\beta) \end{bmatrix},</math>
where appropriate [[List of trigonometric identities#Angle sum and difference identities|trigonometric identities]] are employed for the second equality.
That is, the composition corresponds to the rotation by angle <math>\alpha+\beta</math>, as expected.