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Unformatted text preview: Name: Linear Algebra and Multivariable Calculus Math 20 Fall 2011 Midterm 2 Please write neatly and show all your work , using proper notation. Dont hesitate to ask me questions if anything isnt clear. There are 100 points total. The point values of each question are shown. No calculators. No notes. (1) (5 points each) (a) What is the definition of a vector space? A vector space is a set of vectors V such that if ~v V and ~w V , then both ~v + ~w V and c~v V (and c~w V ) for any c R . In other words, it is a set of vectors closed under linear combinations. (b) What is the definition of a basis for a vector space V ? A basis is a set of linearly independent vectors whose span is V . (c) Give an example of a vector space and a basis for it. There are a lot of possibilities. For example, R 2 , with basis (0 , 1) and (1 , 0). (2) Let A be the following matrix. A = 2 9 1 18 2 (a) (6 points) Find all eigenvalues of A . The characteristic polynomial of this matrix is (2 ) 2 ( 1 ), so the eigenvalues are 2 and 1. (b) (10 points) One of the eigenvalues you found should have been nega tive. For that eigenvalue, the eigenspace has basis 1 . For the other eigenvalue, find a basis for the eigenspace E . We want to find a basis for the eigenspace of the eigenvalue 2. We compute A 2 I , which is 9 3 18 . We want to solve the equation ( A 2 I ) ~x = ~ 0. We can row reduce, which gives the matrix 1 1 3 2 ....
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This note was uploaded on 03/27/2012 for the course MATH 20 taught by Professor Rachelepstein during the Fall '11 term at Harvard.
 Fall '11
 RachelEpstein
 Linear Algebra, Algebra, Multivariable Calculus

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