Some linear operators

Let’s look at the behavior of some linear operators. We’ll work with operators T that are invertible, which means that its range is R^3, that T is surjective: for every vector w, there exists a vector v such that w = T(v). As is true of all linear operators, the origin is fixed since T(0) = 0.

Our program now includes a dropdown menu with some predefined linear operators. When you select an item from this menu, the basis vector fields get populated with the vectors that the basis vectors are mapped to, and the effect of the operator is displayed. For example, for the operator named scale, the basis1 field gets set to 2,0,0 since (for this operator) T(e_1) = (2,0,0). As in the previous program, you can also type into the basis fields directly and press the Go button.

Some linear operators

The following linear operators T essentially transform R^2 in the xy-plane; in other words, T(e_3)=e_3. This allows us to visualize the operator by looking down the z-axis. Each item corresponds to a predefined operator in the dropdown menu:

  • identity: The identity operator maps each basis vector to itself.
  • scale: Stretch by a scale factor of 2 along the x-axis and 3 along the y-axis. Change the values 2 or 3 to stretch (greater than 1) or shrink (between 0 and 1) by different scale factors.
  • reflection: Create a mirror image across the yz-plane. To reflect, scale using the scale factor -1.
  • rotation90, rotation45: Rotate counter-clockwise around the z-axis by 90 or 45 degrees.
  • shear: Translate along the x-axis by an amount equal to distance from the y-axis. Where m is the first value in the basis2 field, lines parallel to the y-axis map to lines with slope 1/m.

Here are a few other operators that function in three dimensions:1

  • rotationA90, rotationA45: Rotate counter-clockwise around the vector (1,1,1) by 90 or 45 degrees.
  • reflectionA: Reflect across the plane x+y+z=0.
  • shearA45: Translate along the z-axis by an amount equal to distance to the xy-plane. The angle between the z-axis and the transformed z-axis is equal to 45 degrees.


  1. You can use the Wolfram Alpha site to compute the matrices for many linear maps. The basis fields in my program correspond to the column vectors of the 3\times 3 matrix A where we use Av to transform vector v. ↩