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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Recall that the function f:X to Y is invertible with inverse function g: Y to X if g(f(x))=x for all x in X and f(g(x))=x for all x in Y. Suppose T: R^n to^n is a matrix transformation defined by T(x)=A(x). Prove that if A is an invertible matrix then T is an invertible function. Hint: Can you think of another matrix transformation that acts as the inverse for T?
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Suppose that f : R? → R² is a linear transformation and that [i]= ([:), ([:) -[:]. ([:)-[4] f f Find a matrix A such that the following equation is true for any vector x: Af(x) = x (In other words, find the matrix associated to the inverse of the linear transformation f. The matrix associated to f satisfies the equation f(x) = Ax instead of the equation above.)
2. Let d≥ 1, let F be a field and let Affa(F) denote the set of all functions TA, FdFd which have the form T₁,v(x) = Ax + v, where A is an invertible d x d matrix with entries in F and where v Fd. Prove that Affa(F) is a group (where the binary operation is given by composition of functions).
a b 1. Let T be a group of all invertible 2 ×2 matrices of the form 1 x Where a,b,c €R and ac#0. Let u be the set of matrices of the form 0 1 where χER. a. Prove that U is a subgroup of T . b. Determine whether U is a normal subgroup of T.
Find the first 5 terms of the solution: (x+2)y' - 3y = 1+x+x² ; Xo=0
. Linear transformation T: Rm →R" is given by one of the matrices below. Find m and n. 1 (a) [ 1 4 ] [ (b) 4 -2 3 8 i] 0 11 (c) 2-2 3 -3
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.
We throw a symmetrical dice twice. Let X denote the number of throws in which an odd number of dice fell, and let Y denote the remainder of the product of the dice's rolls divided by 3.Check whether the variables X and Y are (a) independent (b) uncorrelated
Suppose that T is a linear transformation such that 8 ¹ (H) = [$], ¹(A) = [] T Write T as a matrix transformation. For any z E R², the linear transformation T is given by T() x. =
For a linear transformation L : R² → R³ is L(2, 1) = (2, 6, 8) and L(0, 1) = (2,0, 4). a) What is L(1,0)? b) What is the matrix representation of L? c) Use the matrix representation to calculate L(3,-1).

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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).