Affine transformations by Matrix

Affine transformations of the plane:

The group of matrices of the form below with ad-bc≠0 forms the affine group of the plane.

The matrix M

The linear action of such matrices preserves the plane z=1 in R3. We identify this plane with the (x,y)-plane in the obvious way. So, the matrix acts according to the following equation:

The matrix M

Multiplication by M converts from the new coordinate system to the old coordinate system.

Example: Reflecting Text

In the following example, we reflect the text "Hello!" in the red line, y=0.

image of a circle
Your browser did not display the SVG image. It is showing a standard image as a fallback.
Source of transform-text.svg:Hide line numbers
01: <svg version="1.1" baseProfile="full"
02:      xmlns=""       
03:      width="300px" height="110px" viewBox="0 -55 300 110"
04:      font-family="Times, serif" font-size="50px">
06:    <text x="5" y="-5">Hello!</text>
07:    <line x1="0" y1="0" x2="300" y2="0" stroke="red" stroke-width="1px"/>
08:    <text x="5" y="-5" transform="matrix(1,0,0,-1,0,0)" fill="blue">Hello!</text>
10: </svg>


This presentation is part of a SVG Tutorial for Mathematics Students.

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