代做MAST10007 Linear Algebra, Semester 1 2024 Written Assignment 5代写留学生数据结构程序

School of Mathematics and Statistics

MAST10007 Linear Algebra, Semester 1 2024

Written Assignment 5

Submit a single pdf file of your assignment solutions via the MAST10007 website

before 12 noon on Monday May 6.

• This assignment is worth 2% of your final MAST10007 mark.

• Assignments must be neatly handwritten, but this includes digitally handwritten documents using an ipad or a tablet and stylus, which have then been saved as a pdf.

• Full working must be shown in your solutions.

• Direct questions by email to Jean-Emile, who will post answers to frequently asked questions on the Ed Board.

• Part of your overall mark is for quality of exposition.

Note: You can find fully worked solutions from a previous year on Canvas.

• You are encouraged to discuss the assignment with your friends in small groups but need to write up the solution in your own words.

• Paying for help with the assignment or using Chegg or other discussion groups with a large number of participants constitutes academic misconduct.

Question 1: Change of reference frame. in special relativity

This exercise can be solved using only the material seen in class, but we advise you to watch the special lecture on Special Relativity on Canvas which explains where the change of basis formulas come from. You are also encouraged to search on wikipedia if you want to know more about some of the phenomena described here.

In special relativity, events are assigned a vector of dimension four  corresponding to their coordinates (x, y, z) in space and their recorded time t. This is called an event vector. The choice of basis for these vectors is called a reference frame.

(a) Aristotle is sitting on a rock, he has a lantern blinking every second. He uses the standard basis of R4, namely  for his reference frame. Assuming the first blink happened at the origin at time t = 0 (i.e. event vector ), what is the event vector of the second blink? (units in seconds).

(b) Blaise is passing by on a very high speed motorbike, at speed v measured as a fraction of the speed of light (see the picture). Special relativity tells us that his reference frame. has the modified basis  where  (the scalar γ is called the Lorentz factor).

(i) What is the matrix PAB for the change basis from B to A?

(ii) Deduce the matrix PBA for the change of basis from A to B.

(iii) What do you observe? (Hint: Your expression for PBA can be simplified using the expression of γ given above.)

(iv) What are the event vectors for the first and second blink in Blaise’s reference frame?

(v) Deduce the time between the two blinks measured in Blaise’s reference frame. Notice that γ > 1 and explain the effect of speed on time periods in special relativity.

(vi) What is the distance traveled by Blaise between the two blinks as observed by Aris-totle?

(vii) Blaise deduces the distance he travels from the position of the lantern in his reference frame. What is the distance he measures?

(c) Aristotle now positions his lantern at a point in space with arbitrary coordinates (x, y, z). He records the first blink at t = 0.

(i) What are the new event vectors for the first and second blink in Aristotle’s reference frame?

(ii) What are the coordinates of these vectors in Blaise’s reference frame.

(iii) What is the time between the two blinks measured in Blaise’s reference frame? What do you conclude?

(d) For Blaise to accurately measure the frequency of the Lantern, he should take into account the distance traveled between the two blinks, this is the Doppler effect. The adjusted time period between two blinks is the measured period plus/minus the distance traveled during this time (in our specific units). We take the convention that v > 0 if Blaise is going away from Aristotle (see picture).

(i) Knowing that frequency is just one over the (adjusted) time period, use your result from b) vii) to show that the frequency observed by Blaise is

where fA = 1 is the frequency in Aristotle’s frame. (1 Hz corresponds to one blink per second).

(ii) The same variation of frequency is observed for light. In the table below are the frequencies corresponding to various colors of the visible spectrum. Aristotle’s lantern is green, but Blaise sees it yellow. Is Blaise moving closer or further away from Aristotle? What is Blaise’s speed (approximatively)?

Note: It was observed in 1912-1914 that the frequency of the light emitted by galaxies is shifted, in this case toward red colors. This is called the “redshift of galaxies”. It led to the conclusion that galaxies are moving away from Earth, and that the universe is expanding.

Question 2: Quantum gates

(a) Logic gates perform. manipulations on bits which are series of 0 and 1, i.e. set of elements of F2. When the output is a single bit, they can be seen as a function of vectors in F2. For instance, the elementary logic gates AND, XOR and NOT are given in the example below.

Which of these functions are linear transformations? Justify you answer.

(b) Instead of manipulating bits, quantum circuits use quantum bits, or Qbits, which are vectors in C2. Indeed, the standard basis of C 2 is identified with the quantum states , but now all linear combinations  with (α, β) ∈ C2 are allowed. Quantum gates can often be represented as matrices acting on these Qbits, and we will study here several examples. The simplest example is the NOT gate (also called Pauli X gate) acting on a single Qbit,

(i) What is the rank of this matrix? Is it invertible?

(ii) What is the action of MNOT on the standard basis? Explain why it is called the NOT gate.

(iii) Find two vectors  and  forming a basis of C 2 and such that MNOT = and MNOT = −.

(iv) Let B = {, } be the ordered basis defined by these two vectors, compute the matrices PBS and PSB corresponding to the change of basis with standard basis S = {, }.

(v) Explain why we have

(c) Quantum measurement depends on a choice of reference corresponding to a vector  = . It is represented by the following matrix,

Recall that  denotes the complex conjugate of α, and .

(i) Compute . What do you observe?

(ii) Assuming , what is the rank of ?

(iii) Compute  and ? What is the mathematical nature of the quantum measurement?

(d) To describe quantum gates with two or more Qbits, we need to introduce the tensor product. The tensor product of two Qbits can be described as a function F : C2×C2 → C4 ,

The vectors of the standard basis of C4 correspond to the following quantum states

To study this function, let’s fix the first argument to  and define

(i) Show that  is a linear transformation.

Note: we can show in a similar way that the function obtained by fixing the second argument is also a linear transformation. These two properties mean that the tensor product is bilinear.

(ii) Determine the Null space and the Image of for the following choices of vector :

(e) Using the tensor product, we can define quantum gates acting on 2 Qbits as 4×4 matrices. The gates called CNOT and SWAP are defined by the following matrices,

Are these matrices invertible? Study the action of these gates on the standard basis and justify their name.