Prove that each of the following subsets H of M2(R) is a subgroup of the group G of all invertible matrices in M2(R) under multiplication. {: {: :] 1 a a -b а. Н - b. H = a + b a

Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [40] E = 0 0 02 0 D = 0 -1 0 0 0 0.5 [201] F = 10 3 01 (a) Rewrite T: R" → Rm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in R™) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.

Determine whether the graphs G1, G2 with the following adjacency matrices are isomorphic: 0 10 00 1 0 1 1 0 0 0 1 0 1 0 0 A1 = A2 0 1 1 0 10 0 0 0 1 0 1 1000 10 = 0 1 0 1 1 0 100101 0 0 0 0 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1000 01

How large should n be to guarantee that the Trapezoidal Rule approximation to x - 12x- 48x² + 2x + 5) dæ is accurate to within 0.1. n = How large should n be to guarantee that the Simpsons Rule approximation to 2 1(-a - 12a - 48z? + 2x + 5) dx is accurate to within 0.1. n = Hint: Remember your answers should be a whole numbers, and Simpson's Rule requires even values for n Question Help: Video 1 D Video 2 Submit Question P Type here to search hp

Find the Jordan canonical forms for the following matrices [1 1 07 (a) A= |0_1 0 . . 1 (b( A= |0_1 [1 1 07 1 1 . . [1 0 (c) A= |0_0 -1 0 1 [1 1 (d) A = |0 1 . . ܘܘܦ - . [1 0 07 (e) A= |0 1 1 . . -1 [10 07 (f) A= |0 0 1 0 1 0 .

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

Suppose that B = S-¹AS for some n x n matrices A, B and some n x n invertible matrix S. a. Show that if is in ker B, then Sa is in ker A. b. Show that the linear transformation T (*) = S from ker B to ker A is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B.

{[ ] andez}: a Let R = 0 d ||a, b, EZ} be the subring of M2 (Z) of upper triangular matrices. Define the map ¢ : R → Z × Z by • ([: :)- a = (a, d) 0 d (1) Show that o is a homomorphism. (2) Show that o is onto. (3) Find ker(ø) (4) Use the First Isomorphism Theorem to find what familiar ring R/ ker(o) is isomorphic to.