Chapter 3: Linear Transformations

MA 237

Richard Hitt

Matrix Transformations

We can watch what happens to the unit square in under matrix transformations.  First, we need to describe the unit square parametrically.

In order to keep track of what gets transformed where, it would be better to color code the lines.

Multiplication of Matrices

We can set Mathematica so that it displays matrices in a more human readable form using the following trick.

So if we enter a matrix like

the result is displayed in matrix form rather than as a list of lists.

We can rotate the square by any given angle about the x-axis using the following matrix transformation.

So, for example, if we use a specific angle, we get

We can define

We can spin the square around the x-axis by graphing many different images as θ varies from 0 to 2π.  This can be somewhat automated by loading an animation package.

This package contains an Animate command that we can use to assemble the different graphs in a table.  If you then select the entire table of output by clicking the mouse on the grouping bar on the right, Control-y (or Cell->Animate Selected Graphics) will begin the animation.  The animation controls are in the lower left of the window.

Images

We will need the following package.

You may find it useful to use the AppendRows or AppendColumns commands in the MatrixManipulation package.

We can generate random numbers in the range 0 to 1 using the Random[] function (the range can be adjusted if desired).  So putting three of these Random commands together will generate random vectors in 3-space.  Then we can form their images my multiplying by M.  Since the list will be long, you can suppress the output using a semi-colon at the end.

This shows that if you take points out of the unit cube in 3-space, their images under M all lie on a straight line going through the origin in the plane.  
Notice that M has rank 1 and a 2-dimensional null space.  These facts can be illustrated as follows.

Write the first column of M as a row vector

The rows of this matrix form a basis for the nullspace of M, so we can depict random linear combinations of the rows to get a graphical representation of the nullspace.

Inverses

The inverse of a square matrix can be computed with the Inverse command.  We begin with a 5x5 matrix built from random integers selected between -10 and 10.

The inverse is:

If you prefer a decimal approximation, try

To check that we have the inverse, we can compute:

Solving Linear Systems

Here is a review of the different methods we have for solving linear systems in the context of Mathematica commands.  Much of this material comes directly from the Mathematica help facility.We begin with a 2x2 coefficient matrix.  Much of this material comes directly from the Mathematica help facility.

m = {{1, 5}, {2, 1}}

m . {x, y} == {a, b}

Solve[ %, {x, y} ]

LinearSolve[m, {a, b}]

Inverse[m] . {a, b}

RowReduce[{{1, 5, a}, {2, 1, b}}]

m = {{1, 2}, {1, 2}}

LinearSolve[m, {a, b}]

NullSpace[ m ]

m . %[[1]]

m = {{a, b, c}, {2 a, 2 b, 2 c}, {3 a, 3 b, 3 c}}

NullSpace[ m ]

LinearSolve[{{1}, {1}}, {1, 0}]

m = {{1, 3, 4}, {2, 1, 3}}

v = LinearSolve[m, {1, 1}]

NullSpace[m]

m . (v + 4 %[[1]])

Exercises

Here are the four On Line exercises from section 3.4 in the text translated in Mathematica.

Exercise 1

Let A be the matrix from exercise 2a on page 177.  Use  LinearSolve to solve the system A.X=B where

Exercise 2

Read the exercise in the text on page 182 and solve it in Mathematica.

Exercise 3

Again, read the problem and solve in Mathematica.  Be sure to enter the matrix using fractions rather than decimals to maintain the exactness in the calculations.

Exercise 4

In this problem, you may find it useful to measure the magnitude of a vector (also called the norm).  You can build a function for doing this.

Now you can work the problem.