Math 260, Fall 2018, 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.

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

1. $n\times m$ system of linear equations.

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?

--------------------End of assignment--------------------

Solutions:

Definitions:

1. An $n\times m$ system of linear equations is a system of the form $$\begin{align*}a_{1,1}x_1+a_{1,2}x_2+\dots+a_{1,m}x_m &= b_1 \\ &\vdots \\ a_{n,1}x_1+a_{n,2}x_2+\dots+a_{n,m}x_m &= b_n\end{align*}$$ where the $a_{i,j}$ and $b_j$ are fixed constants, and the $x_i$ are variables.