Math 260, Fall 2015, Assignment 2

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

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

1. Row-equivalent (matrices).
2. (In) reduced row-echelon form.
3. Reduced row-echelon form (of a given matrix).
4. Leading one (of a row-echelon matrix).
5. Leading variable (of a system of linear equations).
6. Addition (of geometric vectors).
7. Scalar multiplication (of geometric vectors).
8. Linear combination (of geometric vectors).
9. Addition (of algebraic vectors, or of matrices).
10. Scalar multiplication (of algebraic vectors, or of matrices).
11. Linear combination (of algebraic vectors, or of matrices).
12. Product (of two matrices).

Carefully describe the following algorithms:

1. Gauss-Jordan elimination.

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

1. Uniqueness of the reduced row-echelon form.

Solve the following problems:

1. Section 1.2, problems 1, 3, 5, 7, 18, and 34.
2. Section 1.3, problems 1, 13, 14, 15, and 34.

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