MATH 0014: Algebra 3 – Further Linear Algebra 2021-2022

联系我们： 手动添加方式: 微信>添加朋友>企业微信联系人>13262280223 或者 QQ: 1483266981

MATH 0014: Algebra 3 – Further Linear Algebra

2021-2022

Exercise Set

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

f (x) = 4×3 + 28×2 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 justi cation, 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 、