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M235A1 - MATH 235 Review of Linear Algebra I Assignment 1...

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MATH 235 Review of Linear Algebra I Assignment 1 Hand in questions 2,4,7,8,9,12 by 9:30 am on Wednesday September 20, 2006. 1. Solve the following system of equations for x and y : x 2 + xy - y 2 = 1 2 x 2 - xy + 3 y 2 = 13 x 2 + 3 xy + 2 y 2 = 0 2. Solve A x = b and y A = c where A = 1 - 1 - 1 2 - 1 - 3 1 0 - 2 , x = x 1 x 2 x 3 , b = 2 6 4 , y = ( y 1 , y 2 , y 3 ) , c = (0 , - 2 , 2) . 3. A system of linear equations in x 1 , x 2 , x 3 has the augmented matrix 1 0 1 b 1 1 0 a 1 a b a , where a and b are real numbers. Determine values of a and b , if they exist, such that the system has: (a) no solution; (b) a unique solution; (c) infinitely many solutions. For the values of a and b that you find in part (c) above, give the general solution to the system of equations. 4. Find A - 1 for A = 1 i 2 2 0 6 - i 1 - i . 5. Determine if the following set of vectors in C 3 is linearly independent: v 1 = (1 , 0 , - i ) , v 2 = (1 + i, 1 , 1 - 2 i ) , v 3 = (0 , i, 2). 6. Determine whether the following subsets W are subspaces of V : (a) V = M 33 , W = { A M 33 : A is nonsingular } ; (b) V = M nn , W = { A M
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