Math 260, Fall 2017, Assignment 3

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

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

1. $n$-component algebraic vector.
2. $\mathbb{R}^n$
3. Sum (of two algebraic vectors).
4. Scalar multiple (of an algebraic vector).
5. Dot product (of two algebraic vectors).
6. Length (of an algebraic vector).
7. Angle (between two algebraic vectors).
8. Orthogonal (algebraic vectors).
9. Product (of two matrices).
10. Matrix form of a linear system.
11. Identity matrix.
12. Inverse (of a matrix).
13. Sum (of two matrices).
14. Zero matrix.
15. Scalar multiple (of a matrix).

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

1. Theorem concerning multiplication by an identity matrix.
2. Theorem concerning commutativity of matrix multiplication.
3. Theorem concerning associativity of matrix multiplication.
4. Distributive laws (concerning matrix operations).

Solve the following problems:

1. Section 1.3, problems 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 34, and 35.
2. Section 2.3, problems 1, 3, 5, 7, 13, 33, 34, 35, 36, 37, and 43 (in problems 33-43, you may ignore the instructions concerning geometric interpretation, for the present).
3. Section 5.1, problems 1, 3, 5, 6, 7, 8, 9, and 10.

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