Question

In: Advanced Math

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if...

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if and only if the points z1, z2, z3, z4 lie on a line or a circle. Then, compute the cross ratio of 1+√ 3, 1−3i, −1 − i and 1 + i and determine whether they lie on a line, a circle, or neither.

Solutions

Expert Solution


Related Solutions

(a) Let U={(z1, z2, z3, z4, z5)∈C: 6z1=z2, z3+ 2z4+ 3z5= 0}. Check if U is...
(a) Let U={(z1, z2, z3, z4, z5)∈C: 6z1=z2, z3+ 2z4+ 3z5= 0}. Check if U is a subspace. If U is a subspace, then find a basis which span U.
(a) look at these the complex numbers z1 = − √ 3 + i and z2...
(a) look at these the complex numbers z1 = − √ 3 + i and z2 = 3cis(π/4). write the following complex numbers in polar form, writing your answers in principal argument: i. z1 ii. z1/|z1|. Additionally, convert only this answer into Cartesian form. iii. z1z2 iv. z2/z1 v. (z1) -3 vi. All complex numbers w that satisfy w 3 = z1. (b) On an Argand diagram, sketch the subset S of the complex plane defined by S = {z...
If the statement is true, prove it. Otherwise give a counter example. a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C},...
If the statement is true, prove it. Otherwise give a counter example. a)If V=C3 and W1={(z1,z2,z2)∈C3:z1,z2∈C}, W2={(0,z,0)∈C3:z∈C}, then V=W1⊕W2. b)If Vis a vector space and W1, W2 are subspaces of V, then W1∪W2 is also a subspace of V. c)If T:V→V is a linear operator, then Ker(T) and Range(T) are invariant under T. d)Let T:V→V be a linear operator. If Ker(T)∩Range(T) ={0}, then V=Ker(T)⊕Range(T). e)If T1,T2:V→V are linear operators such that T1T2=T2T1, and λ2 is an eigenvalue of T2, then...
given z1=2+j3, z2=7+j1, z3=5+j9. find the value of the following expressions in both coordinate and Euler...
given z1=2+j3, z2=7+j1, z3=5+j9. find the value of the following expressions in both coordinate and Euler forms. a) z1z2,z2z3,z3z3(dash on top z),(z1+z3)z2 b) z1/z2,z2/z3,z3(dash on to[ z)/z3,(z1+z2)/z3.
1. How many complex numbers z are there such that z3 = 1? 2. Translate x(t)...
1. How many complex numbers z are there such that z3 = 1? 2. Translate x(t) = -cos(πt + π/3) into standard form A⋅sin(2πft + φ) (There are multiple correct answers) 3. If fs = 100 Hz, what are three aliasing frequencies for f = 80 Hz? 4. A signal x is delayed by one sample and scaled by −1/2, producing a new signal y[n] = −1 2 x[n − 1]. (a) How does Y[m] relate to X[m]? (b) What...
PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers
  PROOFS: 1. State the prove The Density Theorem for Rational Numbers 2. Prove that irrational numbers are dense in the set of real numbers 3. Prove that rational numbers are countable 4. Prove that real numbers are uncountable 5. Prove that square root of 2 is irrational
Let A be a subset of all Real Numbers. Prove that A is closed and bounded...
Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...
Prove the following statements by using the definition of convergence for sequences of real numbers. a)...
Prove the following statements by using the definition of convergence for sequences of real numbers. a) If {cn} is a sequence of real numbers and {cn} converges to 1 then {1/(cn+1)} converges to 1/2 b) If {an} and {bn} are sequences of real numbers and {an} converges A and {bn} converges to B and B is not equal to 0 then {an/bn} converges to A/B
Prove the following formulas, where u, v, z are complex numbers and z = x +iy....
Prove the following formulas, where u, v, z are complex numbers and z = x +iy. a. sin(u+v) = sin u cos v + cos u sin v. b. cos(u+v) + cos u cos v - sin u sin v. c. sin^2 z + cos^2 z = 1. d. cos(iy) = cosh y, sin (iy) = i sinh y. e. cos z = cos x cosh y - i sin x sinh y. f. sin z = sin x cosh...
Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only...
Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only if every infinite subset of E has a point of accumulation that belongs to E. Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real numbers is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E. Must use Theorem 4.21 to prove Corollary 4.22 and there should...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT