Math 260, Spring 2017, Assignment 5

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

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

1. Linear transformation (from $\mathbb{R}^m$ to $\mathbb{R}^n$).
2. Matrix (representing a transformation).

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

1. Theorem concerning representability of linear transformations by matrices.
2. Formula for the matrix representing a linear transformation.
3. Formula for the orthogonal decomposition of a vector with respect to a line.

Solve the following problems:

1. Section 2.1, problems 1, 3, 5, 16, 17, 19, 25, 27, and 29.
2. Section 2.2, problems 5 and 26 (you will need to skim the section for the definitions of "reflection," "scaling," and "shearing," and for the matrices that represent these transformations).

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