Linear Transformations of Points



Click and drag the red and orange arrows to change the matrix `A` of the linear transformation.
(Alternatively, you can set the matrix by using the text boxes below the applet.)
You can show or hide points on the screen by using the checkboxes.
Drag the points `u`, `v`, `w`... around to see how their images `Au`, `Av`, `Aw`... change.


  1. Move `u` and `v` around and see how `Au` and `Av` change. Confirm that the point `Au` is really the result of applying the matrix `A` to the point `u`.
  2. Put several points on the screen and place them in a straight line. What do you see?
  3. Which transformation makes `Au = u`, `Av = v`, `Aw = w`, etc.?
  4. Find a transformation that reflects points across the `x`-axis. Same for `y`-axis.
  5. Find a transformation that multiplies vectors by `2`. So, `Au` = `2u`, `Av` = `2v`, etc.
  6. Find a transformation that projects points "orthogonally" onto the `x`-axis. So, for example `A(3, 5) = (3, 0)` and `A(-13, -7) = (-13, 0)`.  More generally, `A(x, y) = (x, 0)`.
  7. In the previous example, what is the range of the transformation?  The codomain?  What are their dimensions?
  8. Find a matrix `A` so that `Au` always lies somewhere on the line `y = x`, no matter where `u` is.
  9. Place points `u`, `v`, `w` on the screen so that `u` + `v = w`. What do you notice about `Au`, `Av`, `Aw` ?  Does this relationship hold for all linear transformations?
  10. Place `u` at the point `(1, 0)`, place `v` at the point `(0, 1)`, and then try a few different transformations. How are `Au` and `Av` related to the columns of `A`? Can you prove this relationship?


Note for Windows users:
Ctrl + [click and drag] on the background to move the coordinate axes around.
Ctrl + [scroll wheel] to zoom in and out.


Marc Renault, Created with GeoGebra