# Transformation Matrix

Aug 29 2019, 9:35 AM PST 2 min

Martrix transformation is an advanced math concept. This article is meant primarily as a reference, rather than a teaching guide. To teach yourself how to perform matrix transformations, we recommend Khan Academy's lessons on Linear Algebra.

## Simple multiplication

A transformation matrix can be thought of as containing the three new coordinate axes in its columns, since transforming the coordinate axes results in a vector matching the contents of a column:

 a d g b e h c f i
×
1
0
0
=
a
b
c
 a d g b e h c f i
×
0
1
0
=
d
e
f

 a d g b e h c f i
×
0
0
1
=
g
h
i

## Identity

The identity matrix is a transformation matrix that maps every point onto itself (i.e. transforming by it has no effect)

I =

 1 0 0 0 1 0 0 0 1

## Scaling

S(x, y, z) =

 x 0 0 0 y 0 0 0 z

## Rotation

### X-axis rotation

Rx(θ) =

 1 0 0 0 cos θ -sin θ 0 sin θ cos θ

### Y-axis rotation

Ry(θ) =

 cos θ 0 sin θ 0 1 0 -sin θ 0 cos θ

### Z-axis rotation

Rz(θ) =

 cos θ -sin θ 0 sin θ cos θ 0 0 0 1
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