1. Find which pairs of the following groups are isomorphic:(i) The dihedral group D3 of symmetries of an equilateral triangle.(ii) The group of (complex) matrices1001ω 0010 ω0ω 000{(69) -(69) (69) (18)-(25)·(25)}under matrix multiplication, where w = C and w³cube root of unity.)(iii) ZX={1, 2, 4, 5, 7, 8} under multiplication modulo 9.(iv) Z2 × Z3.=1,w 1. (That is, w is a particular

Question 1:Step 1:Step 2:Step 3:

Answered: 1. Find which pairs of the following…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.CR: Review Exercises

Problem 29CR: Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices...

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An operator T on a complex inner product space is normal if TT - TT. State the Complex Spectral Theorem b) Which of the following matrices can be diagonalized over C²? (14) (3-3)
213. Find the entries of the following matrices: (a) the 2 x 2 matrix M for the reflection across the line y = x. (b)the 2 × 2 matrix N for the 90-degree counterclockwise rotation about the origin. (c) the product MN; what transformation does this represent? (d) the product NM; what transformation does this represent? (e) the product MM; what transformation does this represent?
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Do NOT row reduce!(a) Write the augmented matrix whose row reduction would find [x]B .(b) Write the augmented matrix whose row reduction would find [x]C .(c) Write the augmented matrix whose row reduction would give the linear transformationthat takes [x]B to [x]C ..
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 that 0 is 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).
Consider the ordered bases B₁ = {(2,5), (3, 7)} and B₂ = {(−1, 2), (1, −4)} of R². If I is the identity map on R² (that is, the map defined by Iv = v for all v € R²), find: (a) The matrix of I with respect to B₁ for the domain and B₂ for the range (b) The matrix of I with respect to B₂ for the domain and B₁ for the range For the two matrices you found in the previous problem, show that they are inverses of one another.
2. (a) What is the 2 x 2 matrix M that reflects v = = ($) across the x axis, i.e. takes ? Find the inverse matrix of M and explain why your answer X (") to ( ²₁ ) ? Y for M-1 makes sense geometrically. (V³). - (+) across the line l, by using matrix multiplication to compose M from part (a) with a rotation by an appropriate angle 0. (b) Let & be the line through the origin (0, 0) that is parallel to the vector Find the 2x2 matrix N that reflects v = (c) Find N-1 in two ways: by applying the usual formula for the inverse of a 2 x 2 matrix to N, and by using the rule (AB)-¹ = B-¹A-¹ and your representation of N as a product from part (b).
2(b) Use matrix multiplication to find the reflection of (2, -6, 3) about (i) the xy -plane and (ii) the yz -plane. Show the appropriate graph for the problem.
d) A square matrix with the characteristic polynomial λ5 − 3λ4 − 2λ3 + 5λ2 − 4λ + 2 is invertible (attached in image). True or false. e) If A and B are similar then both of them are diagonalizable. True or false.
Let GL(2,11) be the group of all invertible 2 × 2 matrices with entries in Z₁1, with group operation given my matrix multiplication. Consider the following two matrices in this group (where an entry listed as k is shorthand for [k]11): 3 3 10 4- (1₂0). B- (B) A = B= 8 8 (ii) Consider the subset of GL(2,11) given by G= {Am B : m, n = Z}. Show that G is a subgroup of GL(2, 11).
(2) Find the examples that satisfy the statement in the given items and show some work how they are correct examples (a) A 4 x 4 matrices A that is not digonalizable. (b) (c) Two matrices A and B that have the same characteristic polynomial but are not similar. Two diagonal matrices D₁ and D₂ that are similar but they are not equal. (d) Two similar matrices A and B such that A is orthogonally digonalizable but B is not orthogonally diagonalizable.

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