# Coordinates

1. Let be an ordered basis for where

What are the coordinates of in the ordered basis ?
2. Let and be vectors in such that

and

Show that is a basis for . Find the coordinates of in this ordered basis. What do the conditions mean geometrically?
3. Consider the matrix

Show that is invertible. Conclude that represents a transformation of coordinates in . What is the geometric interpretation of the change of coordinates represented by ?
4. Let be a vector space over the complex numbers of all functions from to , i.e., the space of all complex valued functions on the real line. Let , and .
1. Prove that are linearly independent.
2. Let , and . Find an invertible matrix such that

5. Let be the real vector space of all polynomial functions from into of degree 2 or less. Let be a fixed number and define

Prove that is a basis for . If

what are the coordinates of in this ordered basis ?
Subhashis Banerjee 2009-09-03