Spring 2000 final exam

Linear Algebra with Applications

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Math 21b Final Exam - Spring 2000 This was a three hour exam. Total points = 90. (1) (12 pts) Let 1 234 5 6 78 9 1 0 1 1 12 --  =   A . (a) (2 pts) Find rref( A ), the reduced row echelon form of A . (b) (4 pts) Find bases for ker( A ) and image( A ). (c) (6 pts) Find an orthonormal basis for ker( A ). Now let 0 2 4 2 = v . Show that v ker( A ) and express v in terms of your orthonormal basis for ker( A ). (2) (10 pts) Let 0 001 0 010 0 100 1 000 = - - J . We say that a 4 × 4 matrix A is symplectic if AJA T = J . (a) (2 pts) Show that if A is symplectic, then A -1 is symplectic. (b) (2 pts) Show that if A and B are symplectic, then AB is symplectic. (c) (2 pts) Show that J itself is symplectic. (d) (4 pts) Given v , w R 4 , define < v , w = v T Jw . Is this an inner product on R 4 ? Why or why not? (3) (10 pts) Find all solutions to the differential equation 3 ( )2 ( ) ( )4 t f t f t f te ′′ - += . Find the unique solution given the initial conditions (0 )1 f = and (0 f = . (4) (8 pts) Let V = { B M n ( R ): B + B T = 0 } be a set of real n × n matrices. (Recall that such matrices are called anti-symmetric .) Show that V
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This homework help was uploaded on 01/23/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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