2. Let G be a group. Prove that the identity element e is unique. Prove that for-1 with aaa € G, there exists a unique inverse element a= e = a ¹a.3. Let GL2 (R) be the group of invertible 2 x 2 matrices over R.(45)(a) Show that A =(b) Compute A-¹.is in GL₂ (R).

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Answered: 2. Let & be a group. Prove that the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. a. Let 7,(x, y, z)=(y.-x+5y+z, 7z, y-2z) be a linear transformation. Find the matrix of T₁. b. Let 0 23 -27 3 be the matrix of 7₂. Find the linear transformation 7₂.
15. Show your step-by-step solution and answer.
2. Let T: R³ → R² be given by 3x1 + 2x2 4.x2 – 5x3 T I2 and let L: R² → R³ be given by (:)- 6x1 + 12 -3.12 (a) Show that T and L are linear transformations by showing that they are given by matrix multiplication. (b) Find the standard matrix for the composition Lo T.
3. 3. Let be de fined by the line an transformation 2y+7 X - 4y 3x I accordling matrix of the base of opace find the 1. 110 3.
1. The group SU (2) is the group of special unitary (2 x 2) matrices, i.e., the unitary (2 x 2) matrices with determinant 1. A representation of the elements of SU (2) is Û (a) = exp(-ã.5) where a Ŝ € R, j = 1,2,3 and 3 = is the spin operator which is proportional to the vector of Pauli matrices & = (₁, 6₂, 63)¹ 02, o 0 1 and 63 = (1₁) 0 2₁ = (₁ 2)₁ 6₂ = (₁ 3²1) 1), ¹) and 1 0 a. Write [Û (a)] as a linear combination of 2 × 2 Pauli matrices to show that Û (a) = cos ½ – i sin ªª. ☎ 22
) Let G SIL(2, R) be the group of all 2 x 2 matrices with determinant 1. Let Z(G) = {2 € G | zr = #z for all rE G). Find Z(G).
4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.
2. Let H be the set of 2 × 2 matrices of the form H a b = { (- / + d ) | a,b ≤ c } a where a denotes the complex conjugate of a complex number a. Show that H, together with matrix addition and multiplication, forms a ring. (You may assume that M₂(C) is a ring.)
The group of matrices with determinant is a subgroup of the group of invertible matrices under multiplication. а. 3 O b. -1 С. C. 4 d. 1 Ое. 2
3.5.2.f

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2. Let & be a group. Prove that the identity element e is unique. Prove that for a € G, there exists a unique inverse element a¹ with aa-¹ = e = a ¹a. 3. Let GL2 (R) be the group of invertible 2 x 2 matrices over R. (45) (a) Show that A = (b) Compute A-¹. is in GL₂ (R).