finalsol 211 - Stony Brook University Introduction to...

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Unformatted text preview: Stony Brook University Introduction to Linear Algebra Mathematics Department MAT 211 Julia Viro May 15, 2008 Solutions for the Final Exam Problem 1. Solve the system using the Gauss-Jordan elimination. Verify your solution. x 1 + x 2- x 4 + x 5 = 0 2 x 1 + 2 x 2 + 2 x 3 + 2 x 4 + 5 x 5 = 6- x 1- x 2 + x 3 + 3 x 4 + x 5 = 3 3 x 1 + 3 x 2- 3 x 4 + 2 x 5 = 0 Solution. Elementary row transformations of the augmented matrix of the system give rise to the reduced row-echelon form: 1 1 0- 1 1 2 2 2 2 5 6- 1- 1 1 3 1 3 3 3 0- 3 2 1 1 0- 1 1 0 0 2 4 3 6 0 0 1 2 2 3 0 0 0- 1 1 1 0- 1 0 0 0 2 4 0 6 0 0 1 2 0 3 0 0 0 0 1 1 1 0- 1 0 0 0 1 2 0 3 0 0 0 0 1 0 0 0 0 0 . The number of free variables in the solution is the number of unknowns minus the rank of the matrix, that is 5- 3 = 2. Let x 2 = t and x 4 = s . Then x 1 =- x 2 + x 4 =- t + s and x 3 = 3- 2 x 4 = 3- 2 s . Obviously, x 5 = 0. The solution is ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) = (- t + s, t, 3- 2 s, s, 0) , t,s R . Let us verify (check) the solution: - t + s + t- s + 0 = 0 2(- t + s ) + 2 t + 2(3- 2 s ) + 2 s + 0 = 6- (- t + s )- t + 3- 2 s + 3 s + 0 = 3 3(- t + s ) + 3 t- 3 s + 0 = 0 . All equations of the system are satisfied for all values of t and s . So the solution is correct. Answer: ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) = (- t + s, t, 3- 2 s, s, 0) , t,s R . 2 Problem 2. a) Show that skew symmetric 2 2 matrices form a subspace of the space M 2 of all 2 2 matrices. Find a basis and the dimension of this subspace. b) Define the inner product in M 2 by < A,B > = tr( A T B ) for all A, B M 2 . Find a basis in the orthogonal complement of the subspace of skew symmetric matrices. Solution. a) A skew symmetric matrix A satisfy the condition A T =- A . Let A, B be arbitrary skew symmetric matrices and k be a real number. Then ( A + B ) T = A T + B T =- A- B =- ( A + B ) , which means that A + B is skew symmetric, and ( kA ) T = kA T = k (- A ) =- ( kA ) , which means that...
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finalsol 211 - Stony Brook University Introduction to...

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