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The final exam questions for a mathematics 205 course, focusing on linear transformations and orbits. The exam covers topics such as onto and one-to-one mappings, determinants, eigenvalues, and kepler's laws of planetary motion. Students are required to show their work and solve problems involving matrices, vectors, and differential equations.
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Mathematics 205 Final Exam: C December 16, 2010
Problem Possible Actual 1 10 2 10 3 20 4 15 5 25 6 20 (5) Total 100
You must show all work to receive credit. No electronic devices other than calculators are permitted. Give exact answers (such as ln 5 or e^2 ) unless requested otherwise.
(a) What does it mean for a mapping to be onto? Is T (~x) = A~x onto?
(b) What does it mean for a mapping to be one-to-one? Is T (~x) = A~x one-to-one?
(a) What are the two properties of a linear transformation?
(b) Suppose that ~u 6 = ~v. Does it follow that T (~u) 6 = T (~v)? Justify your answer.
(c) Is it necessarily the case that T (~0) = ~0? Justify your answer.
(d) Suppose that ~u and ~v are orthogonal. Does it follow that T (~u) and T (~v) are orthogonal? Justify your answer.
(3, .88), (2. 3 , 1 .1), (1. 65 , 1 .42), (1. 25 , 1 .77), (1. 01 , 2 .14).
Using a least-squares approximation, determine the type of orbit. If e < 1 the orbit will be an ellipse. If e = 1 the orbit will be parabolic. If e > 1 the orbit will be hyperbolic.
(a) Show that the map T : D → F defined by T (f ) = e−tf ′(t) is a linear transformation.
(b) What property would eigenvectors of this transformation satisfy? (Hint: it is a differential equa- tion).
(c) Bonus: Find the eigenvectors and eigenvalues.
