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Unformatted text preview: Math 54 Final Review You can get a pdf of this review sheet from my website: <http://math.berkeley.edu/ ∼ asmarks> . Solutions to most of the practice problems on this sheet will also eventually be posted at my website. The final will be on Wednesday December 17, in the same room (150 Wheeler) that lecture is in. The exam will cover material since the second midterm. You should expect the questions on the exam to be very similar in difficulty and style to those on the homeworks. Some strategies for test taking For problems with equal point values, work on the easiest problems first. Don’t get stuck on a single question when there are others you could be doing much more quickly. Read all the questions carefully – you don’t want to lose points for misreading a question. Show all your work. This may increase the amount of partial credit you get, and helps you make fewer mistakes. Explain what you are doing if you use a theorem that is nonobvious. It’s worth taking an extra minute to write up your answers well. Check your work near the end of the exam to make sure that you’ve have answered each question entirely, especially multipart questions. If you have time, check your work for logical and arithmetic mistakes. If you still have time, check your work in any way you can. Solve problems a different way, draw pictures, do special cases, etc. You shouldn’t leave the exam until the last possible moment. Symmetric matrices (Lay Chapter 7) • A symmetric matrix is a matrix A such that A = A T . The symmetric matrices are precisely the matrices that are orthogonally diagonalizable. A matrix A is orthogonally diagonalizable iff A can be diagonalized with an orthogonal basis of unit eigenvectors. That is, A = PDP 1 = PDP T where the columns of P are orthogonal unit eigenvectors, and D is a diagonal matrix whose diagonal entries are the eigenvalues of A . To find an orthogonal set of eigenvectors, you must use the GrahamSchmidt process on a basis for each eigenspace. All the eigenvalues of a symmetric matrices must be real (so if you get imaginary eigenvalues, you’ve made a mistake). In problems where you are asked to orthogonally diagonalize a matrix, you can check your work by making sure that AP = PD , or A = PDP T . • A quadratic form is a function Q : R n → R of the form Q ( x ) = x T A x , where A is an n × n symmetric matrix. If we make a change of variables of the form x = P y , then Q ( x ) = ( P y ) T A ( P y ) = y T P T AP y . If P is a matrix that orthogonally diagonalizes A , so that A = PDP T , then y T P T AP y = y T D y ....
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 Spring '10
 Aho
 Microeconomics, Fourier Series, Partial differential equation, Hilbert space, general solution, T T

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