Exercise 3.2.13. Let H(R) a - {(1; i)=ancex} = a, b, 0 0 1 (1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group. (2) Find an explicit example of matrices A, B € H(R) such that AB ‡ BA. (3) Is H(R) a subset of GL3(R)?

In the following problems, decide if the groups G and G are isomorphic. If they are not, give properties of the two groups that show there can be no isomorphism from G onto G. If they are isomorphic, provide an explicit isomorphism. (a) G = GL(2, R), the group of 2 × 2 nonsingular matrices under multiplication; G = (R − 0, ·), the nonzero real numbers under multiplication. (b) G = (R, +), the real numbers under addition; G = (Q, +), the rational numbers under addition (c) G = Q4, the group of quaternions; G = D4, the dihedral group of symmetries of the square

Let Z denote the set of integers, and let 1 0 G 0 1 0 0 |a Z} 0 1 Prove that G together with the usual matrix multiplication forms a group

2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?

Which of the following matrices are invertible? (Select all of them.) 0 (88) 0 0 1 0 (8 5) 0 The 3 x 3 identity matrix. The 2 x 2 matrix corresponding to counterclockwise rotation by π/3.

In the multiplicative group of invertible matrices in M4(R), find the order of the given element A. 1 1 b. A = 1