Page 1 of 5 SIT787 -Mathematics for Al Assignment 2 Related to the content of Weeks 2-5 Trimester 2, 2024 Important notes: e Submission deadline can be found on the unit website. Your submission can be handwritten but it must be legible. If your submission is not legible, it will not be marked and will result in a zero mark. A proper way of presenting your solutions is part of the assessment. Additional rules may be applied to specific questions. If you cannot follow these rules for justifiable reasons, you must contact the unit chair for possible alternatives. o Please follow the order of questions in your submission. o All steps (workings) to arrive at the answer must be clearly shown. No marks will be awarded for answers without workings. o Generally, you need to keep your answers in the form of quotients and surds (e.g. % and v/3). Rarely, you may convert your solutions into decimals for plotting or comparing purposes. However, you need to show the final answer in terms of quotients and surds before converting them into decimals. o Only electronic submission would be accepted via the unit website. Your submission must be in ONE pdf file. Multiple files and/or in different file format, e.g. .Jpg, will NOT be accepted. Tt is your responsibility to ensure your file is not corrupted and can be read by a standard pdf viewer. Failure to comply will result in a zero mark. We have a zero-tolerance policy for academic misconduct, such as collusion. If your work is accused of academic misconduct and later substantiated, you will receive no marks for the entire assignment. SIT787 Mathematics for Al Assignment 2 2024 Tri 2 Page 2 of 5 Question 1) Consider these vectors: , and w = o = = O [ =) (i) Determine which two vectors are most similar to each other based on these norms: (a) f2 norm: mn dist(x,y) =[x ylla = ly ll2 =, | > (2 3:)? i=1 (b) norm: n dist(z,y) = |z yll1 = Iy 2l = 5 loi i i=1 (ii) Determine which two vectors are most similar to each other based on the cosine similarity measure: Ty cos(f) = [z[[[ |yl [You can use decimals here. However, you need to show the final answer in terms of quotients and surds before converting them into decimals.] (iii) Explain the reason behind the difference in result between (i) and (ii) if you observe any (Word limit is 80 words). (iv) Can the difference be resolved? Give details of your suggestion, if you have any, and explain the outcome if your suggestions are applied (Word limit is 80 words). No marks will be given if you DO NOT follow these rules: e For (iii) and (iv), type your answer using a computer rather than hand writing. This is the only way your marking tutor can mark your statement without misunderstanding. e Any words exceeding the word limit will not be considered for marking. [6+5+4+5+5= 20 marks] Question 2) You are tasked with uncovering information about an incomplete matrix, some of whose entries are unknown and denoted as a,b,, and d : A= o o = RO N _ = o n R (i) Find the rank of A based on the values a,b, , and d. (ii) Given @ = 1,b = 2, = 3, find all the singular values in terms of d. (You may wish to solve Question 3 related to singular values before solving this one; you still need to follow the order of questions in your submission. Equivalently, you can search for the definition of singular values.) \fSIT787 Mathematics for AI Assignment 2 2024 Tri 2 Page 4 of 5 Question 3) Consider the following matrix. 0 1 (i) Find the full solution set for Ax = X E R 3 (ii) Find rank(A), a basis for the column space of A, C(A), a basis for the row space of A, C(AT), a basis for the null space of A, N(A), and a basis for the left null space of A, N(AT ). Hint: For a m x n matrix A, the column space of A is defined as C(A) = { Ax/x E R"} C Rm, and the row space of A is defined as C(AT) = {Alyly E Rm} C R". Also, the nullspace N(A) is defined as {x E R"|Ax = 0} C R". For the same matrix, the left nullspace N(AT ) is defined as {y E Rm|ATy = 0} C Rm. Also, we have dim(C(A)) + dim(N(AT)) = m dim(C(AT)) + dim(N(A)) = n. (iii) Construct B= AA, and find eigenvalues and eigenvectors of B. For positive eigenvalues of B, define oi = vli, where A1 2 12. Construct matrix D as follows. D = 0 02 0 (iv) If the eigenvectors of B are not orthonormal, orthonormalise them. Make a matrix U using the orthonormal vectors you obtained. The ordering of the columns of U should be the same as the ordering of the eigenvalues, that is U = [uj u2]. In other words, u1 is the eigenvector corresponding Aj and u2 is the eigenvector corresponding to 12. Show that U is an orthogonal matrix. (v) Find eigenvalues and eigenvectors of the matrix G = A'A, and orthonormalise them. Call them v1, v2, and v3 according to the order of A1 2 12 2 13 of eigenvalues of G. (vi) Orthogonalise { v1, v2, v3}, and construct the matrix V = [v1 v2 v3], and check whether it is an orthogonal matrix. (vii) Find three vectors {w1, w2, w3 } so that Wi = LATui, i = 1,2 oi W3 1 W1, W3 I w2, and | |wall = 1. Compare {w1, w2, w3} with {v1, v2, v3} and comment similar characteristic of these sets. (viii) Compute the product UDVT. (ix) Compare the columns of U and V with the bases you found for C(A), C(A"), N(A), and N(A" ). What can you say about the columns of U and V in terms of the bases for the four subspaces of the matrix A? [No marks will be given if you DO NOT follow the following rules: (1) Word limit is 80 words. Any words exceeding the word limit will not be considered for marking. (2) Type your answer using a computer rather than hand writing. This is the only way your marking tutor can mark your statement without misunderstanding.]SIT787 Mathematics for AT Assignment 2 2024 Tri 2 Page 5 of 5 [5-+5+5+5-+5+5+5+5+5= 45 marks] Question 4) Select one particular question from this assignment, i.e., Question 1, Question 2, or Question 3; explain how the question selected is related to Al. Read the Rubric for this question on the unit website. Exemplar: I selected question X, which is about vector calculations. In AT applications, information is typically processed by computers that excel at manipulating high-dimensional vectors. By manipulating vectors, an Al system can extract patterns and insights from data that may not be readily apparent in the raw inputs. Through various vector operations and transformations, these systems can provide valuable analysis and understanding of complex datasets, enabling them to perform tasks such as classification, prediction, and pattern recognition. No marks will be given if you DO NOT follow these rules: e Type yvour answer using a computer rather than hand writing. This is the only way your marking tutor can mark your statement without misunderstanding. e Word limit is 100 words. Any words exceeding the limit will not be considered for marking,. [20 marks]