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MATH 0014: Algebra 3 – Further Linear Algebra

2021-2022

Exercise Set

1) Consider the following polynomials in R[x]:

f (x) = 4x3 + 28x2 − 55x + 9    ,    g(x) = x2 + 7x − 14

By applying the Euclidean algorithm to f (x) and g(x):

a) determine the greatest common divisor of f (x) and g(x), gcd(f, g);

b) determine polynomials a(x) and b(x) in R[x] such that gcd(f, g) = a(x)f (x) + b(x)g(x).

2) Consider the following matrices:

╱  、           ╱  、           ╱  、

( 0      0     −7．     ,     ( 0      0     −9．     ,     ( 0      0     −7．

For each of these matrices, determine, with justification, the minimal polynomial of the matrix.

3) Consider the following complex matrix:

M  =   ╱、

(0   0      3  ．

Determine distinct invertible matrices P and Q, and a matrix J in Jordan normal form, such that P − 1 MP = J and Q− 1 MQ = J.

4) For each of the following matrices, determine the real and complex canonical form(s) of the matrix:

、     ,     3   3   −、