讲解：MATH 240、MATLAB、MATLAB、formatProlog|R

MATH 240 Spring 2019 MATLAB Project 3 – due in class Tuesday, 4/9Directions:There are no new commands to learn for this project. Please review old commands as needed.Previous guidelines on format and collaboration hold. Please review them if you forget them. For thisproject, do problem 1 in format short and do the rest in format rat.As before, a question part marked with a star indicates the answer should be typed into your outputas a comment – the question isn’t asking for MATLAB output.0. Review all directions and rules from the previous project. Then enter the command clock.1. Do #36 on p. 226 of the textbook.2. (Use format short) Consider the set of functionsWe can view this set as a set of vectors in the vector space C(R) of all continuous real-valued functionswhose domain is R. We would like to show that this is a linearly independent set in C(R). This meansthat if x1, x2, x3, x4 are scalars such that() x1(1) + x2 cost + x3 cos2t + x4 cos3t = 0 (for all t),then we must have x1 = 0, x2 = 0, x3 = 0, x4 = 0. Essentially we are trying to disprove the existenceof a linear relation relating these four functions to each other. More concretely, we are trying to provethat no identity such as3 + 4 cost 7 cos2�could possibly exist.(a) Each substitution of a number for t in the equation (?) produces a linear equation in the fourvariables x1, x2, x3, x4. By plugging in t = 0, 0.1, 0.2 and 0.3, you get four linear equations for thefour unknowns. Define the coefficient matrix A for this linear system in MATLAB.(b) Note that a nontrivial solution x to (?) is automatically a nontrivial solution to Ax = 0. However,if A is invertible, then Ax = 0 has no nontrivial solutions. This implies that the equation () hasno nontrivial solutions. Compute rref(A) and det(A).(c) Very briefly explain why each of the last two computations show A is invertible.(d) It is reasonable to be suspicious of the very small value of det(A) in the last step – could this beroundoff error, with the actual det(A) being zero? Do a check by repeating the computation withthe more spread out inputs t = 0, .2, .5, 1, to see det(A) large enough to eliminate that suspicion.(e) Now consider the set of functions {1,sin2t, cos2t}. (Here 1 denotes the constant function whosevalue is always 1.) Explain why this set of functions is linearly dependent. (Hint: Do you knowany identities that relate these three functions?)(f) (Optional - don’t ha代做MATH 240作业、代做MATLAB编程设计作业、MATLAB实验作业代写、代做format留学生作业 代做留学生ve to turn in) Explore what happens if you try to use the techniques ofthe previous parts on the set {1,sin2t, cos2t}.3. Do #34 on p. 232 of the textbook.4. (Use format rat) Let A.(a) Compute rank A. (The MATLAB command is rank(A).)(b) Use the rank to determine the values dim(Nul A), dim(Col A), and dim(Row A).(c) Compute rref(A) and use it to give a basis fori. Nul A ii. Col A iii. Row A5. Consider the polynomialsp1(t) = 3 + 5t + 5t3, p2(t) = 1 + t + 2t�which are all elements of the vector space P3. We shall investigate the subspaceW = Span{p1(t), p2(t), p3(t), p4(t), p5(t)}.(a) Let vi = [pi(t)]E , the coordinate vector of pi(t) relative to the basis E = {1, t, t2, t3} for P3. Enterthese coordinate vectors into MATLAB as v1, v2, v3, v4, v5.(b) Let A be the matrixv1 v2 v3 v4 v5. Observe that Span{v1, v2, v3, v4, v5} = Col(A). Usethis fact to compute a basis for Span{v1, v2, v3, v4, v5}. (Recall you can enter A into MATLABas A = [v1 v2 v3 v4 v5].)(c) Translate your previous answer into a basis for W (consisting of polynomials). What is dim W?(d) Is W = P3? Justify your answer.6. Consider the following four matrices from the vector space M2×3 of all 2 × 3 matrices:�(a) Let vi denote the coordinate vector [Ai]E relative to the basisEnter the coordinate vectors for A1, A2, A3, A4 into MATLAB.(b) Use MATLAB to show that the coordinate vectors v1, v2, v3, v4 are linearly dependent.(c) ? Express one of the matrices Ai as a linear combination of the other three. (Hint: first do thesame for the coordinate vectors.)7. (Optional - You do not have to turn this problem in. It is based on material in section4.7, which we will not finish until too close to the due date of this project.)Let [x]B denote the coordinate vector of x with respect to a basis B. For bases B and C, P C←Bdenotesthe change of coordinates matrix, which has the property that PC←B[x]B = [x]C. It follows thatAlso, if we have three bases B, C, and D, then�Each of the following three sets is a basis for the vector space P3:E = {1, t, t2, t3} ,B = {1, 1 + 2t, 2 t + 3t2, 4 t + t3} , andC = {1 + 3t + t3, 2 + t, 3t t2 + 4t3, 3t} .(a) Compute the matrices P = PE←Band Q = PE←C.(b) Use P and Q and the properties above to compute R = PC←B.(c) Compute the C coordinate vector of the polynomial t3.(d) Suppose p(t) is the polynomial for which [p(t)]B . Compute the coordinate vector [p(t)]C.转自：http://www.7daixie.com/2019041013143597.html