linearTransformation
Jack Picciuto
Frank Wattenberg
Department of Mathematical Sciences
United States Military Academy
West Point, NY 10996
Jack.Picciuto@usma.edu
Frank.Wattenberg@usma.edu
Linear Transformations |
Linear Systems |
The osslet linearTransformation is a flexible and easily used osslet that can be used to investigate linear transformations
of the form:
Quick Examples
- When you click on the link below "A first example" you will see a 2D graph
of the x-y plane. You will also see a blue ball
and red ball
along
with a black square ■.
The blue ball is the only ball you can move around by clicking and dragging it
around the grid. You will notice that the red ball moves as you move the
blue ball around. The blue ball is your x-vector while the red ball is the
transformation of this vector -- we are calling it the y-vector here.
Notice that as you move the blue ball, the numerical display showing the
coordinates of the two vectors x and y is automatically updated.
- The transformation matrix is the A-matrix. The values (which you can
edit) determine the transformation that the x-vector will undergo.
- The black square represents the b-vector that is used in curriculum units exploring a system of equations Ax = b. You can change the location of this vector by editing the numerical entries for the b-vector
in the usual way.
- Your goal is to move the x-vector (blue ball) around the grid while
watching the movement of the red ball. You are trying to get the red ball
to land directly on top of the black box. When you do this, you have now found
the appropriate x-vector that the A-matrix transforms into the desired b-vector.
The values of the x-vector solve the Ax = b system of equations.
- You are now
ready to try this by clicking on
A first example: This example illustrates some of the capabilities of the osslet linearTransformation.
- Another example:
This is an example of a live graph that might be used for exercises in which
students discover facts about a linear transformation. Notice that the display
of the matrix A is omitted, since students must determine A.
Example Curriculum Units
A Variation
A variation, linearSystem, of this osslet can be used to investigate linear systems of the form:
A Sample Curriculum Unit Using linearSystem
-
Finding Solutions to Initial Value Problems. This curriculum unit explores the solutions for systems of linear equations in which the equilibrium point is attracting, repelling, or a saddle point. It also looks at a system with an infinite number of equilibrium points.
Resources for Curriculum Developers