Can I get an explanation of problem 5c-5e in Ma
thlab.
Use format short. We first show the set of vectors B = {1, cos(T), cos? (T), cos(), cos' (t)} is linearly independent over the vector space of real-valued functions. That is, we want to show the equation 20(1) + Q1 cos(x) + a2 cos? (x) + az cos (x) + 24 cos" (r) = (1) is true only when 2o = 21 = (n = 2z = 24 = 0 (where denotes the zero function that maps everything to 0). Note that (1) must hold for all values of r. If the functions were linearly dependent, then for any dependent relation, say 3-4 cos(1) +12 cos (1) - 2 cos (?) +2 cos' (t) = 0 (this is not a true equation), we should still observe linear dependency for explicit values of r e.g. if we plugged in r = n/23. (a) By subbing in a value for r in (1), we get a linear equation in terms of variables 20, 21, 22, 23, and a4. Sub in r = 0.1,0.2, 0.3, 0.4, 0.5 to create a linear system of 5 equations and 5 unknowns. Define its coefficient matrix by A in Matlab. (b) Observe the system has a non-trivial solution if and only if (1) would have a non- trivial solution. Use appropriate Matlab commands to conclude the system only has a trivial solution, thus implying B is a linearly independent set of vectors. (C) Let C = {1, cos(x),cos(2x), cos(3-7), cos(4x)}. We have the following identities: cos(2x) = -1 + 2 cos' (:) COS(3.c) = -3 cos(x) + 4 cos (1) cos(4x) = 1 - 8 cos (x) + 8 cos(x). With the help of the identities, determine the B-coordinate vectors for the 5 vectors in C. Define these as column vectors in Matlab as u1, u2,... (d) Using part (c), define an appropriate matrix B and use Matlab commands to determine if C is a linearly independent set or not. Use disp or fprintf to briefly explain your answer. (e) If D = Span(B), explain why C forms a basis for D. Use format short. We first show the set of vectors B = {1, cos(T), cos? (T), cos(), cos' (t)} is linearly independent over the vector space of real-valued functions. That is, we want to show the equation 20(1) + Q1 cos(x) + a2 cos? (x) + az cos (x) + 24 cos" (r) = (1) is true only when 2o = 21 = (n = 2z = 24 = 0 (where denotes the zero function that maps everything to 0). Note that (1) must hold for all values of r. If the functions were linearly dependent, then for any dependent relation, say 3-4 cos(1) +12 cos (1) - 2 cos (?) +2 cos' (t) = 0 (this is not a true equation), we should still observe linear dependency for explicit values of r e.g. if we plugged in r = n/23. (a) By subbing in a value for r in (1), we get a linear equation in terms of variables 20, 21, 22, 23, and a4. Sub in r = 0.1,0.2, 0.3, 0.4, 0.5 to create a linear system of 5 equations and 5 unknowns. Define its coefficient matrix by A in Matlab. (b) Observe the system has a non-trivial solution if and only if (1) would have a non- trivial solution. Use appropriate Matlab commands to conclude the system only has a trivial solution, thus implying B is a linearly independent set of vectors. (C) Let C = {1, cos(x),cos(2x), cos(3-7), cos(4x)}. We have the following identities: cos(2x) = -1 + 2 cos' (:) COS(3.c) = -3 cos(x) + 4 cos (1) cos(4x) = 1 - 8 cos (x) + 8 cos(x). With the help of the identities, determine the B-coordinate vectors for the 5 vectors in C. Define these as column vectors in Matlab as u1, u2,... (d) Using part (c), define an appropriate matrix B and use Matlab commands to determine if C is a linearly independent set or not. Use disp or fprintf to briefly explain your answer. (e) If D = Span(B), explain why C forms a basis for D
