Math 220 - February 8, 1999

Discussion

• When sizes match, the multiplication of matrices is associative.

• If f(x) = Ax and g(y) = By, with x a column of length n, A a matrix of size m \times n, and B a matrix of size p \times m, then g(f(x)) = Cx where C is the p \times n matrix product C = BA.

Assignment for Wednesday, February 10

1. Let f be the transformation of the plane given by f(x) = Ax where A is the matrix

   (

   {1}/{SQRT{2}}	{1}/{SQRT{2}}
   -{1}/{SQRT{2}}	{1}/{SQRT{2}}

   ) .

   1. Plot f(x) for several of the points x = ( cos theta, sin theta) on the unit circle.

   2. Describe what f does to every point on the unit circle.

   3. Observing that every point in the plane can be represented as rx where r > 0 is a scalar and x is a point on the unit circle, describe f as a motion of the plane.

2. Let A and B be the matrices

   A = (

   1	1
   0	1

   ) and B = (

   0	1
   -1	0

   ) .

   Compute the two matrix products AB and BA.

3. Let A and B be the matrices

   A = (

   cos a	- sin a
   sin a	cos a

   ) and B = (

   cos b	- sin b
   sin b	cos b

   ) .

   Compute the matrix product AB.

4. Let f be the transformation of space given by f(x) = Ux where U is the matrix

   (

   0	0	1
   0	1	0
   1	0	0

   ) .

   Describe f as a motion of space.