Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MAT301 Assignment 1

You can use all the results from the lectures, tutorials, and the results from the book in that were covered in class before the assignment due date. When you are using a result state clearly and explicitly what you are using. You can give references to theorems in the book or say that a particular statement you are using was proved in class.

You may NOT use any results that were not covered in class. This includes future results from the book that have not been covered in yet. Show work in all problems.

(1) Let D be the following string of ellipses (infinite in both directions). Describe all symmetries of D. Is the group of symmetries of D abelian? 

(2) Find all elements x G D4 such that (DXHR270)²³ = V. Show that you have found all possible solutions.

(3) Let X G D4 such that X = R270. Find all Y G D4 such that XY² = R90. Explain your reasoning.

(4) Let G be the set of 2x2 real matrices A with both eigenvalues having absolute value equal to 1.

Is G a group with respect to matrix multiplication?

(5) Let A be the matrix

/

A = I 0 1 0 I

\-1 0 0 丿

Verify that A e GL(3, R) and compute its order in GL(3, R).

(The order |g| of an element g e G is the smallest positive integer n such that gⁿ = e. If no such n exists then we say that |g| = oo.)

(6) Which of the following maps are symmetries of the specified D? Explain your reasoning.

(a) D = [0,1], f (x) = x³;

(b) D = {x e R, 0 <y < 1}, f (x, y) = (x + 1,1 - y);

(7) Using the Euclidean algorithm find the multiplicative inverse of 253 mod 600 in ZgQQ.

(8) Let Mnxn(Zm) be the set of n x n matrices with entries in

Let A =(爲 1) e M2x2(Zi8)

Does there exist a matrix B e M2x2(Zxs) such that AB =

If yes, find such B, if not, prove that it doesn't exist.