Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
#21. Let G be the set of all real 2 x 2 matrices where ad + 0, Prove that under matrix multiplication. Let N = (a) N is a normal subgroup of G. (b) G/N is abelian.
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
LO 2a 4) Let V be the set of diagonal 2x2 matrices of the form la ). Determine whether or not this set is a subspace of the set of all real-valued 2x2 matrices, M22, with standard matrix addition and scalar multiplication. Justify your answer.
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
please provide with full working solution. thank you
Consider the set B of all 2 x 2 matrices of the form {C 9 b a B a, b e R -b a and let + and . represent the usual matrix addition and multiplication. (a) Determine whether the system B = (B, +,.) is a commutative ring. (b) Determine whether the system B = (B, +, .) is a field. T
Consider the set B of all 2 x 2...
I already solved part A and I just need help with part B
1. Matrix Multiplication The product of two n xn matrices X and Y is a third n x n matrix 2 = XY, with entries 2 - 21; = xixYk x k=1 There are n’ entries to compute, each one at a cost of O(n). The formula implies an algorithm with O(nº) running time. For a long time this was widely believed to be the best running...
16. Let Z(G), the center of G, be the set of elements of G that commute with all elements of G. (a) Find the center of the quaternions, defined in Example 19.16. (b) Find the center of Z5. (c) Show that Z(G) is a subgroup of G. (d) If Z(G) G, what can you say about the group G? b 0 Example 19.16 d We now work inside M2(C), the ring of 2 x 2 matrices with complex entries. Consider...
2. Consider the following set of complex 2 x 2 matrices where i = -1: H = a + bi -c+dil Ic+dia-bi Put B = {1, i, j, k} where = = {[ctdie met di]|1,3,c,dex} 1-[ ), : = [=]. ; = [i -:], « =(: :] . (a) Show that H is a subspace of the real vector space of 2 x 2 matrices with entries from C, that is, show H is closed under matrix addition and multi-...
Q 2 (c) Let S be the set of matrices of the form A = a, a T ag arbitrosy where are real numbers. Show there exists a unique matrix E in s such that АЕА o in S. for all Marks ((1+3+37 +(2+3 + 8) = 20 Marks) MATH 2118 Online Class Exercise I Qla) Sketch the surface s defined by the equation z = =9-6tty! (6) Determine the equation of the tongent plane to the surface s given...