Math 331 - February 8, 1999

Discussion

• Definition. The norm of a point x in R^{n} or of a vector x with n coordinates is denoted ||x|| and defined by the formula

  ||x|| = SQRT{x_{1}^{2} + ... x_{n}^{2}} .

• Definition. The distance d(x, y) between two points x, y in R^{n} is defined by the formula

  d(x, y) = ||x - y|| .

• Definition. The inner product x . y (or scalar product or dot product) of two vectors with n coordinates is defined by the formula

  x . y = SUM_{j=1}^{n}[ x_{j} y_{j} ] .

• Definition. The angle ang(x, y) between two vectors with n coordinates is defined by the formula

  ang(x, y) = arccos ( {x . y}/{ ||x|| ||y|| }) .

• Note. In the Cartesian plane (only) one may also speak of the directed angle from one vector to another.

• Theorem. An affine transformation is distance-preserving if and only if its associated matrix is an orthogonal matrix.

• Theorem. Every distance-preserving transformation is an affine transformation (for which the associated matrix is orthogonal).

• Terminology. Synonyms for ``distance-preserving affine transformation'': congruence, isometry, rigid motion.

Exercises due Wednesday, February 10

1. How many different affine transformations of the plane permute the vertices of a given triangle?

2. How many of the 24 permutations of the vertices of a non-rectangular parallelogram may be realized with an affine transformation?

3. Let f(x) = Ax be the linear transformation of the plane where A is the matrix

   A = {1}/{5} (

   3	4
   4	-3

   ) .

   1. What points x of the plane are ``fixed'' by f, i.e., satisfy f(x) = x ?

   2. What lines in the plane are carried by f to other lines?

   3. What lines L in the plane are ``stabilized'' by f, i.e., satisfy the condition that f(x) is on L if x is on L ?

4. Let O denote the origin, and let P_{1}, ..., P_{n} be any points of R^{m}. Show that the n+1 points O, P_{1}, ..., P_{n} are barycentrically independent if and only if the n points P_{1}, ..., P_{n} are linearly independent.