Math 260, Fall 2019, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 2.3 (note that the material on block matrices and transition matrices is optional).

Carefully define the following terms, then give one example and one non-example of each:

1. Orthogonal (vectors).
2. Addition (of matrices).
3. Scalar multiplication (of matrices).
4. Product (of matrices).
5. Commute (i.e. "two specific matrices $A$ and $B$ are said to commute when...").

Carefully state the following theorems (you do not need to prove them):

1. Properties of the dot product (i.e. valid rewritings of the expressions $\vec{u}\cdot(\vec{v}+\vec{w})$, $(c\vec{u})\cdot\vec{v}$, $\vec{v}\cdot\vec{u}$, and $\vec{v}\cdot\vec{v}$).
2. Law of cosines (in vector form; also known as the "geometric interpretation of the dot product").
3. Formula for the angle between two non-zero vectors.
4. Procedure to determine whether two non-zero vectors make an acute, obtuse, or right angle.
5. Theorem concerning the associativity of matrix multiplication.
6. Theorem concerning the commutativity of matrix multiplication.
7. Distributive laws for matrix multiplication.

Solve the following problems:

1. Section 5.1, problems 4, 5, 6, 7, 8, 9, and 10.
2. Section 2.3, problems 1, 3, 5, 7, 9, 11, 13, and 17. (Hint for 17: if $B$ commutes with $A$, then $B$ must be a $2\times 2$ matrix, so write it as $B=\begin{bmatrix}b_{1,1}&b_{1,2}\\b_{2,1}&b_{2,2}\end{bmatrix}$. Then the condition $AB=BA$ is equivalent to a certain system of linear equations in the four unknowns $b_{1,1},b_{1,2},b_{2,1},b_{2,2}$. You can find all solutions of this system using Gauss-Jordan elimination. The set of all matrices commuting with a given matrix $A$ is called the centralizer of $A$; this exercise shows that the Gauss-Jordan algorithm can be used to compute centralizers. You may wish to try a few more from among problems 18-26 to get a feeling for what centralizers can look like.)

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