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### oldfinal

Course: MATH 103, Fall 2009
School: Stanford
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Word Count: 674

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103 Math Final Tuesday March 20th , 2007 Time: 3 hours Total: 160 points This is a closed book test. Calculators and other computational aids are strictly forbidden. Lucky charms wont help you, but feel free to use them. Cell phones wont help you either, but do not feel free to use them. Good luck 1. The aim of this question is to connect two curves smoothly. Let 1 = {(x, y) | x &lt; 0, y = 0} and 2 = {(x,...

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103 Math Final Tuesday March 20th , 2007 Time: 3 hours Total: 160 points This is a closed book test. Calculators and other computational aids are strictly forbidden. Lucky charms wont help you, but feel free to use them. Cell phones wont help you either, but do not feel free to use them. Good luck 1. The aim of this question is to connect two curves smoothly. Let 1 = {(x, y) | x < 0, y = 0} and 2 = {(x, y) | x > 1, y = 1}. In words, 1 and 2 are two horizontal line segments. 1 starts at the origin and goes left (horizontally). 2 starts at (1, 1) and goes right (horizontally). 10 (a) Let c0 , c1 , c2 , c3 R and dene f (x) = c0 + c1 x + c2 x2 + c3 x3 . Find c0 , . . . c3 so that f (0) = 0, f (1) = 1, f (0) = 0 and f (1) = 0. [If you draw the graph of y = f (x), this will give you a curve that smoothly joins 1 and 2 . For those who forgot calculus, f (x) = c1 + 2c2 x + 3c3 x2 .] 10 10 (b) Let c0 , c1 , c2 R, and dene g(x) = c0 + c1 x + c2 x2 . Show that it is not possible to nd c0 , c1 , c2 so that g(0) = 0, g(1) = 1, g (0) = 0 and g (1) = 0. (c) Suppose you wanted to smoothly join 1 and 2 using a function g as dened in the previous subpart. As youve shown above, an exact solution is not possible. Explain what the next best thing to do would be. Write down an explicit expression for c0 , c1 , c2 obtained in this manner. [Your answer can involve products and inverses of matrices, as long as the matrices consist of only numbers (i.e. no variables).] 2. Let T : R2 R2 be a linear transformation. Let v1 = T v2 = v2 . 10 10 5 5 10 (a) Find the matrix of T and T 1 in basis {v1 , v2 }. (b) Find the of matrix T and T 1 in standard basis. cos sin and v2 = sin cos . Suppose T v1 = v1 and (c) Does the transformation T have any physical meaning? 3. (a) Dene what it means for {v1 , . . . vk } to be linearly independent. (b) Suppose {v1 , v2 , v3 } is linearly independent. Show that {v1 , 2v2 , 3v3 } is linearly independent. 4. Let A = a b c d . Suppose ad bc = 1 and At = A1 . [Your answer can involve a, b, c, d.] 5 10 10 5 10 (a) Write down a formula for A1 . (b) Show that a + d = 0. (c) Compute the characteristic polynomial of A. (d) What are the eigenvalues of A? 5. Let V R4 be dened by [Your answer should not involve a, b, c or d.] 1 1 1 2 V = span , 1 2 0 1 Let x R4 . Find a matrix P so that P x is the orthogonal projection of x onto V . 10 6. Find a matrix P such that 1 P 1 conceptually how P 1 AP relates to A.] 2 3 P = 4 3 1 4 2 [Hint: If you attempt doing this by trial and error, you will waste 3 hours and get ...

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