Math 220 - February 5, 1999

Class Summary

If A is an M \times N matrix and f is the function from R^{N} to R^{M} that is given by

then the following terms apply to f:

Image

The image of f is the subset of the target space R^{M} consisting of all points f(x) as x varies in the domain R^{N}.

Fiber

The fiber of f over y is denoted f^{-1}(y) and is defined, for y in the target R^{M}, as the set of all points x in the domain R^{N} that are carried by f to y, i.e., for which f(x) = y.

Kernel

The kernel of f is its fiber f^{-1}(0) over the origin.

Assignment for Monday, February 8

Please remember that there will be a short test on Monday.

The exercises below pertain to the function f(x) = Ax where the matrix A is given by

1. Find all points that are in the fiber f^{-1}(1, -5, 3).

2. Describe the fiber of the previous exercise as a subset of space.

3. Find equation(s) that characterize the image of f as a subset of space.

4. Describe the image of f as a subset of space.