Question: Prove that every proper affine plane isometry F[x] = Qx + b of R2, where det Q = 1, is either (a) A translation, or

Prove that every proper affine plane isometry F[x] = Qx + b of R2, where det Q = 1, is either
(a) A translation, or
(b) A rotation (7.40) centered at some point c ∈ R2.

Step by Step Solution

★★★★★

3.45 Rating (158 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

If Q I then Fx x a is a translation Otherwise since we a... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (2497).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Prove that every proper affine isometry F(x) = Q x + b of R3, where det Q = +1, is one of the following: (a) A translation x + b, (b) A rotation centered at some point of R3, or (c) A screw...

Let be the line in the direction of the unit vector u through the point a. (a) Write down the formula for the affine map defining the reflection through the line . (b) Write down the formula for the...

Let p be a prime, (a) If G has order 2p, prove that every proper subgroup of G is cyclic, (b) If G has order p2, prove that G has a subgroup of order p.

Let G be a group with G pq where p and q are prime Prove that every proper subgroup of G is cyclic

Question 3 (Matchings; 10 marks) Graph colouring problems, independent sets, and matchings are closely related. A proper vertex colouring is an assignment of colours to the vertices of a graph so...

Let V be a vector space. A subset of the form A = {w + a | w W} where V V is a subspace and a V is a fixed vector is known as an affine subspace of V. (a) Show that the affine subspace A P is a...

You$must$attach$this$cover$page$to$the$front$of$your$HW$or$you$will$lose$5$points.$ $ $ APMA$1860$Homework$2$ $ $ First$Name:$____________________________$$$Last$Name:$______________________________$...

18. 19. 20. 21. 22. 23. 24. 25 26 27 29. 30. 31. 32. 33. 34. 35. Suppose that a is a group element and a6 E e. What are the possi- bilities for lal? Provide reasons for your answer. If a is a group...

19. 20. 21. 22. 23. 24. 25. 26. 27 29. 30. 31. 32. 33. 34. 35. d If a is a group element and a has innite order, prove that a'" as a" when m a n. For any group elements a and b, prove that lab! =...

Given data points (xo, uo). .........(xn un) with x, , i j, prove that there is a unique continuous piecewise affine interpolant u = f(x), so u = 0..........n. (a) Prove that the piecewise affine...

Consider nodal configuration 3 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The boundary is insulated. Explain how Equation...

An economist wanted to investigate the relationship between office rents (the dependent variable) and vacancy rates. Accordingly, he took a random sample of monthly office rents and the percentage of...

BE20.18 (LO 2) On December 31, 2024, Escapee plc leased machinery from Terminator Group for an agreed-upon lease term of 3 years. Escapee agreed to make annual lease payments of 17,000, beginning on...

Consider the graph of the market for cigarettes in problem 4.4, where it is assumed that the government collects the tax from the producers. a How would the graph be different if the tax was...

Consider the syrup dispensing measurements described in Exercise 8-31. Compute a 95% prediction interval on the syrup volume in the next beverage dispensed. Compare the length of the prediction...

Consider the natural frequency of beams described in Exercise 8-32. Compute a 90% prediction interval on the diameter of the natural frequency of the next beam of this type that will be tested....

Compute a 95% tolerance interval on the life of the tires described in Exercise 8-22, that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the...

A put option on a stock with a current price of $37 has an exercise price of $39. The price of the corresponding call option is $2.85. According to put-call parity, if the effective annual risk-free...

Rate of Return If State Occurs State of Probability of Economy State of Economy Stock A Stock B Stock C Boom .20 .34 .46 .50 Normal .40 .25 .23 .20 Bust .40 .03 .25 .42 c - 1 If the expected...

1. Regulation and supervision of the financial system include 3 risk mitigation strategies to balance the safety net, which are: a.Regulation, supervision and enforcement b.Enforcement, supervision...