Math 260, Fall 2017, Assignment 4

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

Read:

1. Section 2.1 (you may skip pages 51-53 if you wish).
2. Section 2.2.
3. Section 2.3 (you may skip the material on block matrices and transition matrices, beginning at the bottom of page 81, if you wish).

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

1. Standard basis vectors.
2. Transformation (from $\mathbb{R}^m$ to $\mathbb{R}^n$).
3. Linear (transformation).

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

1. Theorem concerning linearity of transformations represented by matrices.
2. Theorem concerning representability of linear transformations by matrices.
3. Procedure to find the matrix representing a given linear transformation.
4. Formula for the matrix representing rotation through an angle $\theta$ in $\mathbb{R}^2$.
5. Formula for the parallel and perpendicular components of a vector in $\mathbb{R}^2$, with respect to a given line.
6. Procedure to find the matrix representing orthogonal projection on a given line.
7. Procedure to find the matrix representing reflection in a given line.

Solve the following problems:

1. Section 2.1, problems 1, 2, 3, 4, 5, 6, 16, 17, 18, 19, 20, 21, 22, 23, and 42 (in problems 16-23, you may ignore the part of the instructions concerning invertibility).
2. Section 2.2, problems 2, 5, 10, and 11.
3. Section 2.3, problems 33, 34, 35, 36, 37, 43, 44, 45, and 46 (you have already done problems 33-43 last week, but at that time you ignored the instructions concerning geometric interpretation; now give geometric interpretations of your prior work, referring if necessary to the material on shearing transformations on pages 69-70).

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