MATH3204 Numerical Linear Algebra & Optimisation (S2-2022): Assignment 01
MATH3204 (S2-2022): Assignment 01
1. Vector Spaces [2 marks each]
Determine whether each of the following is a subspace and justify your answer. In all these cases the underlying field is R.
(a) (a,b,c)∈R3 |a+2b+2c=0 (b)(a,b,c)∈R3 |a2 =b
(c) {f ∈ C[0,1] | f(1/2) = 1} with the standard function addition and scalar multi- plication where C[0,1] denotes the set of all continuous functions on the interval
0 ≤ x ≤ 1.
(d) A∈R2×2 |A= 0 a+b , a,b∈R with the standard matrix addition and scalar multiplication.
2. Linear Maps [6 marks each]
(a) Let Pn denote the space of polynomial of degree at most n with real coefficients. Find the matrix representation of the differential operator D : P3 → P2 given by D(at3 + bt2 + ct + d) = 3at2 + 2bt + c with respect to the standard monomial basis for P3 and P2.
(b) Do the same as above with {2, (t + 1)/2, t2} for a basis of the range of D (still use the standard monomial basis for the domain of D).
3. Matrix Range, Rank, and Pseudo-inverse [30 marks]
Let A ∈ C and b ∈ C . Show that the following three statements are equivalent:
• There exists a vector x ∈ Cn such that Ax = b,
• Rank(A) = RankA b where A b ∈ Cm×(n+1) is a matrix obtained by appending b after the last column of A, and • AA†b = b.
4. Spectral Properties [5 marks each]
Supposeu,v∈C , n≥2,u̸=0,v̸=0andletA=uv .
(a) Find all the eigenvalues of A.
(b) When is A diagonalizable and when is it not? In other words, find conditions on u and v that ensure A is not defective.
(c) When is A unitarily diagonalizable and when is it not? In other words, find conditions on u and v that ensure A is normal.
(d) Find all the singular values of A.
5. SVD [5 marks each]
In this question, you will use SVD to compress your image. Take a picture of yourself and load it using a programming language of your choosing. For Matlab and Python, you can respectively look into functions imread and open from Python Imaging Library (PIL).
(a) After loading your image, turn it into gray-scale. Write a short script for computing its truncated SVD. You can use the inbuilt function svd for both Matlab and Python (in the NumPy library). Start with rank r = 2 and go up by powers of 2, to r = 64. Show the resulting images.
(b) Comment on the performance of the truncated SVD. State how much storage is required as a function of r and matrix dimensions and compare it with the storage required for the original picture.
(c) Now do the same but keep the colors, i.e., don’t turn your image into gray-scale. Your code for this part should output colored compressions of your original image that (Hint: consider performing SVD on each RGB color band separately and then combine the results).
6. Positive Semi-definite Matrix [15 marks]
SupposeA∈R ,A =A,and⟨x,Ax⟩≥0, ∀x∈R . Showthat⟨z,Az⟩≥0, ∀z∈
Note:
• This assignment counts for 15% of the total mark for the course. 2
100 marks in total
- Although not mandatory, if you could type up your work, e.g., LaTex, it would be greatly appreciated.
- Show all your work and attach your code and all the plots (if there is a programming question).
- Combine your solutions, all the additional files such as your code and numerical results, all in one single PDF file.
- Please submit your single PDF file on Blackboard.
