Visualizing 2-dimensinal Linear Transformations

The goal of this page is to help you understand how a 2 by 2 matrix defines a transformation of the two-dimensional plane. The wikipedia page on transformation matrices provides a good explanation of how this works. Briefly, each point (x,y) in the plane gets moved to a new point T(x,y) by the simple rule that T(x,y)=Ax, where A is a 2x2 matrix, x is the column vector with coordinates x and y, and we interpret the column vector that results from the matrix multiplication as a new point in the plane. To help visualize this, we can "transform" an image by moving the pixels by this rule (keeping their color and brightness unchanged). You should imagine that the image starts out in the first quadrant of the plane with the origin at its lower left corner.

To use this page, just:
  1) Pick an image. Either one of the two "built-in's" or anything else from the web.
  2) Adjust the entries of the matrix. It starts out as the identity matrix, which won't change the image at all.
  3) Click [try it out]


Choose your Image:
The old TTU math dept logo
The linux penguin from Tuxpaint
Any image from the internet (1 MB max). Paste the url here:

Adjust your Matrix:


Transformed image:


Original image:


Some suggestions (i.e., for my students, homework)

1) Start with diagonal matrices, like

1.5   0.0
0.0   0.5
The above example will stretch the image in the horizontal direction and shrink it vertically. Try putting negative numbers in one or both of the diagonal entries, and explain what happens.

2) Try a matrix with zero's on the diagonal, like

0  1
1  0
The above example swaps the x and y axes, so it reflects the image around the line y=x. You can combine this effect with stretching and shrinking, and you can use negative values to change the reflection to a rotation through a right angle. Be sure you understand what happens in each case, and why.

3) Try a rotation matrix, which has the form

cos(θ) -sin(θ)
sin(θ)  cos(θ)
Pick an angle and get out your calculator to compute the matrix values. Finally, you can use trigonometry for something entertaining!

4) Try more general examples. To predict and understand the effect, just remember that the left matrix column is equal to T(1,0) and tells where the x-axis goes; while the right column equals T(0,1) and tells where the y-axis goes. The rest of the image is just pulled along "linearly". You can tell where a pixel will be by using vector addition:

T(x,y) = x T(1,0) + y T(0,1)

5) Observe how the determinant of the matrix effects the transformed image. If you haven't already, try a matrix whose determinant is zero. This will squash the image to a line (or just a point), so the transformed image will be invisible! What happens when the determinant is almost zero? What does a large determinat do? Note that the total area of the transformed image will be equal to the area of the original times the absolute value of the determinant. If the determinant is negative, then the "orientation" of the image will be reversed. Text will be hard to read, Tux's paintbrush will move from his left hand to his right.

Final note: if your transformed image is a red circle with an x in the middle, it means the program couldn't find your image, or you gave it a matrix it couldn't handle, or maybe you've found a bug in the software.


Please email comments or suggestions to jeff "at" math.tntech.edu.     The "heavy lifting" for this script is done by: Graphics Magick.
The perl code for this page can be viewed: here