Math 260, Fall 2015, 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. Composition (of two linear transformations).
2. Sum (of two linear transformations).
3. Scalar multiple (of a linear transformation).
4. Domain (of a linear transformation).
5. Codomain (of a linear transformation).
6. Image (of a linear transformation).
7. Linear combination (of some set of vectors).
8. Span (of some set of vectors).

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

1. Formula for the matrix representing the composition of two linear transformations.
2. Formula for the matrix representing the sum of two linear transformations.
3. Formula for the matrix representing a scalar multiple of a linear transformation.
4. Associative law (for matrix algebra).
5. Distributive law (for matrix algebra).
6. Theorem characterizing the image of a linear transformation (in terms of the columns of its matrix).

Solve the following problems:

1. Section 2.4, problems 1, 3, 5, 7, 16, 17, 20, 21, and 23.
2. Section 3.1, problems 17, 18, 19, and 20.

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