Math 3323 Matrix Algebra
Investigation 3

Note that when you open up the Mathematica page, it starts with an error message. You may ignore that error messsage; simply hit the Graph button, and you will see the house.

Please modify the instructions as follows:

1. If you were not in class on Thu., Nov. 15, please see me (or a classmate) for hints, including a solution for reflecting about the x-axis.

   Also find the linear transformation that reflects the house about the y-axis.

2. Hint: Example 11 in the textbook.

3. No modification.

4. Hints and discussion: There are only a few (types of) possibilities range and nullspaces of a linear transformation from R^2 to R^2. Each of these is a subspace of R^2, so it must be either: all of R^2 (2-dimensional subspace); the trivial subspace consisting of just the zero vector (the unique 0-dimensional subspace); or a 1-dimensional subspace. (There are infinitely many 1-dimensional subspaces in R^2, but they are still just 1-dimensional.)

   Try to pick linear transformations with the range and nullspace you want. (Hint: If you pick a matrix at random, then almost certainly its range will be all of R^2 and its nullspace will consist of just the zero vector. So you'll have to have a plan to get a different range and nullspace. What's true about a matrix if its nullspace is 1-dimensional? What's true about a matrix if its nullspace is 2-dimensional? What happens to the range in each of these cases? Does there seem to be a relation between the dimension of the range and the dimension of the nullspace?)

   Try to find relations between the range or nullspace, and the effect of the linear transformation on the house. Can you explain any of these relations? This is your main goal for this part (Step 4) of the investigation.

5. Instructions for the report: Write a summary of your observations and conjectures. Describe all the experiments that you did. List all the conjectures you made (including any that later turned out to be false), how you came up with them, how you tested them, and which ones you still think are true. Please use clear, complete sentences, and organize your report. It will probably be best if you organize your report along the lines of the 4 steps listed in the website (modified according to the above). Note that Step 4 will weigh more heavily than the other steps in the grading (approximately double the other steps).