Math 260, Fall 2015, Assignment 4

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

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

1. Dot product (of two vectors).
2. Magnitude (a.k.a length or norm) of a vector.
3. Angle (between two non-zero vectors).
4. Parallel and perpendicular components (of a vector $\vec{v}$ with respect to some fixed vector $\vec{x}$).
5. Inverse (of a linear transformation).

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

1. Cauchy-Schwartz inequality.
2. Formulas for the parallel and perpendicular components of a vector.

Carefully describe the following algorithms:

1. Algorithm to calculate the inverse of a linear transformation (or to establish that it is not invertible).

Solve the following problems:

1. Section 2.2, problems 6, and 10.
2. Section 2.3, problems 1, 3, 5, 7, 9, and 17.

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