Homework #4

1.     For each positive integer n, let Pⁿ denote the linear space of all real polynomials of degree less then n, together with the usual polynomial addition and multiplication by scalars in Ñ.

a)     Show that the polynomials 1, s, and s² form a basis for P³

b)     Show that the function L: P³ ® P² defined by

      is a linear transformation.

c)     Find the representation of L with respect to the bases {1, s, s²} and {1, s} of P³ and P², respectively.

d)     Construct a basis for the kernel of L

2.     Problem 3.14 of Chen’s book.

3.     Problem 3.17 of Chen’s book. There is a typo in the book: the matrix Q should be

4.     Compute the characteristic polynomial, minimal polynomial, and Jordan normal form of the following matrices