Math 260, Fall 2019, Assignment 1

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

Read:

1. Section 1.1.
2. Section 1.2.

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

1. $n\times m$ system of linear equations.
2. $n\times m$ matrix.
3. Augmented matrix (of a system of linear equations).
4. Coefficient matrix (of a system of linear equations).
5. Augmentation column (of a system of linear equations).
6. Elementary row operation (please discuss all three types).

Solve the following problems:

1. Section 1.1, problems 1, 3, 11, 12, 13, 24, and 30.
2. Does it ever happen that a $2\times3$ system of linear equations has exactly one solution? Why or why not? (Hint: the solution set is the intersection of two planes in three-dimensional space. Can two planes ever intersect in a unique point? Try to think about all of the relevant possible relationships among the planes.)
3. Does it ever happen that a $3\times2$ system of linear equations has exactly one solution? Why or why not?

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