Let SL₂(R)= {A|A is a 2x 2 matrix with det(A)=1}. We will show that SL₂(R) is a group with matrix multiplication:(a) Let A,BE SL₂(R). Find det(AB).(b) IS AB E SL₂(R) ?O No, SL₂(R) is not closed under multiplication.Yes, SL₂(R) is closed under multiplication.(c)I=We already know that matrix multiplication is associative and that0is a multiplicative identity for matrix multiplication. Find det(I).01(d) Is IE SL₂(R) ?O No, SL₂(R) does not contain an identity element.O Yes, SL₂(R) contains an identity element.(e) Let AE SL₂(R). Find det(A-¹).(f) Is A-¹ ESL₂(R) ?O Yes, every element of SL2(R) has an inverse in SL₂(R).O No, not every element of SL₂(R) has an inverse in SL₂(R).

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Answered: Let SL₂(R)= {A|A is a 2x 2 matrix with…

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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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.)
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.
Please clearly illustrate your opinion on this question. Thank you.
Let L be the linear transformation from R2 → R² given by L(x1, x2) = (2x1 - 3x2,4x1 + 4x2). (i) What is the matrix A of L? A = (ii) What is the determinant of L? Determinant = (iii) Is L shape preserving? (No answer given) (iv) Does L preserve orientation? (No answer given) (v) Is L injective? (No answer given) (vi) Is La clockwise rotation? (No answer given) (vii) Is L an anti-clockwise rotation? (No answer given) (viii) Compute the pre-image of the point (2, –1) under L.
Chads baseball team is practicing hitting in the batting cages for pitching machines pitch baseballs at the rate of 35 pitches in 5 minutes chad practiced hitting for 8 minutes how many baseballs were pitched to chad in that time
. 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
3x1-x2 (E) 1. Let T : R? → R³ be the linear transformation defined as T 2x1 + x2 %3D -X1 - X2 (a) Find its matrix representation. (b) Find T(0). (c) Find T-(0). 1 (d) Find T-
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.
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.
Consider the linear transformation from R² to R² known as “"rotation by -90 de- grees." The matrix of this transformation is Is this transformation onto ? Is this transformation injective (one-to-one) ? What is the image of 0 1 [99] -1 0 1 2 ? If there is an inverse for this transformation, what is the matrix of the inverse?
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).
Let T: R² R² be defined by T and C Given Pc 3 -{]-[H]} -1223) , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). [T](PB (u)) 7 [4₁]) B-{B] R]} = Let u = = Ex: 5 [-3x1 + 2x₂ / -4

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