Hi expert, please answer the following problems..

Question 2

2 Determine if the statements below are True or False. If it's True, explain why. If it's False explain why not, or simply give an example demonstrating why it's false. A correct choice of "True" or "False" with no explanation will not receive any credit. 2.1 If a 2 x 2 matrix A is non-invertible, then the image of the unit square under the linear transformation TA is a line segment. 2.2 If T : R' - R' is a linear transformation given by counter-clockwise rotation around the origin by some angle 0, then T either has no (real) eigenvalues, or it has eigenvalue 1 (only). 23 If T : R3 - R' is a function which has the property Tax + by) = aT(x) + bT(y) for all a, be R and all x, y e R, then T is a linear transformation. 24 If T : R' - R' is a linear transformation which has T(e, ) = T(ez), then AT has eigenvalue 0.QUESTION 1 (a) Suppose a, b and c are integers with god(a, b) = 1 and suppose alc and blc. Show that (ab)|c. [9 marks] (b) Find integers a, b and c with ale and blc but ab/c. [6 marks] (c) Show that every permutation o in S, is expressible as a product of transpositions. [7 marks] (d) Express the permutation (1, 4, 7)(2, 8, 6,3,5) as a product of transposi- tions. [3 marks]