The due date for this theme is Friday, February 15, at noon.

Reading assignment: Section 2.1, 2.2

Background reading: Section 2.3

In this theme, we will study some algebraic properties of matrices. To begin, you will need some practice adding matrices and multiplying a matrix by a scalar. Solve problems 1 through 4 on page 107. To gain some practice multiplying matrices, do problems 5 and 6 on page 107. Do not hand in solutions to these problems.

Note that the algebraic properties of matrices given in Theorem 1 and Theorem 2 are similar to properties of the real numbers. While matrix multiplication satisfies many of the same properties as the multiplication of real numbers, there are some notable exceptions. For example, Example 7 on page 105 shows that matrix multiplication is not commutative; the order that you multiply the matrices matters. Another surprising omission is the following. If the product of real numbers is 0, then at least one of the numbers must be zero. However, it is possible to multiply two nonzero matrices together and get the zero matrix. Let be the matrix

Find a nonzero matrix such that , where is the matrix with all entries.

Next, we will study the inverse of a matrix. We say that an matrix has an inverse, which we denote , if

Before addressing the problems below, gain some facility finding and using inverse matrices by solving problems 1 through 6 on page 117. Do not hand in these solutions.

Suppose that . Let

Use the ``Algorithm for finding '' on page 116 to derive the formula for in Theorem 4:

For this problem, you must show and explain all steps of your work. (Do not substitute numbers for , , , and in your derivation.)

Let

Use Theorem 6 to find the inverse of the matrix in two ways. Which technique is easier? Why?

Consider the collection of all matrices. Find three elementary matrices, one to switch the second and third rows, another to multiply the first row by 4, and the third to add 3 times the third row to the second row. Now, find the inverses of these three matrices. According to the book, they are elementary matrices. What elementary row operation does each perform?

Suppose that is invertible. Then, the columns of are linearly independent. Explain why this statement is true. (Do not just cite Theorem 8 in Section 2.3.)

We are now ready to work the following problem:

The standard basis vectors in are

Any vector in can be expressed as a linear combination of the basis vectors

In many fields, the ability of computers to display 3-dimensional images and rotate the images in space to see the object from different perspectives is extremely important. Here is a simplified version of how this works. Imagine that the object in its natural orientation is written in terms of the basis ; more specifically, imagine that points to the right, points up, and points directly out of the monitor. To rotate the image, the computer writes each point of the image in terms of a new set of basis vectors and then displays the image as if these new basis vectors point right, up and out of the monitor. Suppose we wish to rotate our object by 45 and then view the picture from the back. The matrix transformation that performs this is

The matrix transformation maps basis vectors to basis vectors. Find the basis vectors determined by ; call them , and . Find the inverse of , , and show that , , and .

Suppose that a brick is represented in standard coordinates by the following eight vectors, one for each vertex of the brick:

Not all the vertices need be visible. Find the images of the eight vertices. Draw pictures of the original brick and the rotated brick, identifying corresponding vertices.