Question 4.1 (15 marks) Suppose that V is a set of vectors such that N 2 N V = -2 ! لما Let W = span(V), and let y = - 5
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Question 4.3 (10 marks) Does the set of vectors CIBE form a basis for R'? Show you argument(s) clearly, Question 44 (10 marks) Consider the following homogeneous system *4 + 3x2 + 3x3 - X4+ 2xy = 0 *: + 2xy + 2xy - 2x + 2x = 0 *1 + x2 + x3 - 3x + 2x = 0 Find a basis for the solution space of the given system and the corresponding dimension Question 4.5 (20 marks) Consider a basis for R$ Use the Gram-Schmidt Orthogonalization Process to find an orthogonal basis for the space
Question 4.6 (15 marks) For each of following, determine whether the set S is a subspace of the vector space V and explain your arguments clearly, (a) V = R, S = {(x,x + 2): x € R). (b) V = R', S = {(x,y,z.x + 2y - z):x,yz E R}, Question 4.7 (10 marks) Given t:R3 - R t(x, y, z) = (x cose - y sind, sino + y cose,x) where is a given angle. Prove that the function is a linear transformation. Question 4.8 (5 marks) Prove that the column space of a square matrix A of size m x m span "if and only if the matrix equation Ax = b has a solution for each bin R".