Question 1) Consider these vectors: u = and w (i) Determine which two vectors are most similar to each other based on these norms: (a) 2 norm: dist(x, y) =||xy||2|3x||2 = (- Y₁)² (b) ₁ norm: n dist(x, y) =||-||-||-||1=-yil i-1 (ii) Determine which two vectors are most similar to each other based on the cosine similarity measure: エリ cos(0)= [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. (iv) Can the difference be resolved? Give details of your suggestion, if you have any, and explain the outcome if your suggestions are applied./nQuestion 2) You are tasked with uncovering information about an incomplete matrix, some of whose entries are unknown and denoted as a, b, c, and d: -1 0 a b 4 d 00 (i) Find the rank of A based on the values a, b, c, and d. (ii) Using the results you obtained in (i), how many distinct eigenvalues does A have considering a, b, c, and d?/nQuestion 3) Consider the following matrix. 1 A= (i) Find the full solution set for Ax= - 8. -1 , XER³ (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 € R"} CR", and the row space of A is defined as C(AT) = {ATyly € R"} CR". Also, the nullspace N(A) is defined as {x = R"|Ax = 0} CR". For the same matrix, the left nullspace N(AT) is defined as {y R" Ay=0} CR". Also, we have dim(C(A)) + dim(N(AT)) = m dim(C(AT)) + dim(N(A)) = n. (iii) Construct B = AAT, and find eigenvalues and eigenvectors of B. For positive eigenvalues of B, define σ = √, where A ≥ A2. Construct matrix D as follows. D= 0 002 (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 = [u u2]. In other words, u₁ is the eigenvector corresponding A₁ and u₂ is the eigenvector corresponding to A₂. Show that U is an orthogonal matrix. (v) Find eigenvalues and eigenvectors of the matrix G=ATA, and orthonormalise them. Call them 1, 2, and 3 according to the order of A₁ ≥ A2 ≥ As of eigenvalues of G. (vi) Orthogonalise (v1, v2, v3), and construct the matrix V-[v1 v2 vs], and check whether it is an orthogonal matrix. (vii) Find three vectors {1, 2, ws) so that σ A-1,2 wwww2, and ||wa||= 1. Compare {1, 2, wa} with {1, 2, 3} 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(AT), N(A), and N(AT). What can you say about the columns of U and V in terms of the bases for the four subspaces of the matrix A?

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