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Unformatted text preview: AMS 510.01 Fall 2007, Midterm #1 Solutions 1. Indicate whether each of the following statements is true or false. n should be taken as any constant positive integer. (2 points each). (a) A system of linear equations may have exactly two solutions. False. A system of linear equations may have one unique solution, no solution, or an infinite number of solutions. (b) For any nonsingular matrices, A and B , if A 1 = B 1 , then A = B True. The matrix inverse is unique for any given (nonsingular) matrix. (c) The dot product of any two complex vectors is a real number, i.e. : ~u ~v R , ~u, ~v C n False . While the dot product of any complex vector with itself is a real, the same is not true for the dot product of two different complex vectors. (d) The set of all integers forms a field. False. A field must contain the multiplicative inverse of every nonzero element in the set. The integers do not satisfy this requirement. (e) The vectors, { (1 , , 0) , (0 , 1 , 1) , (0 , 1 , 1) } form an orthonormal set. False. While the vectors are orthogonal, they are not all normal the second and third vectors are both of length 2. (f) Every nonzero complex number has a unique multiplicative inverse. True. Just as the multiplicative inverse exists and is unique for all nonzero real numbers, the same is true of complex values. (g) The following matrix is nonsingular: 1 1 1 1 1 False. The matrix contains two identical rows (1 and 3) and thus must be singular. (h) The vectors of a basis must be orthogonal to one another. False. All that is required of the vectors of a basis is that they are linearly indepen dent. (i) For any nsquare matrices, A , B and C : [( A + B ) T C ] T = C T A + C T B True. [( A + B ) T C ] T = [( A T + B T ) C ] T = [ A T C + B T C ] T = C T A + C T B (j) Let ~u = (3 4 i, 2 5 + 2 i ) C 2 .  ~u  = 7. True. The norm of a complex vector can be found by taking the square root of the sum of the squares of the real coefficients of each complex value. Thus:  ~u  = q 3 2 + ( 4) 2 + (2 5) 2 + 2 2 = 9 + 16 + 20 + 4 = 49 = 7. 2. Consider the set of three vectors in C 3 , S = { ~u, ~v, ~w } , where: ~u = (1 i, , i ) ~v = (0 , 1 + i, 0) ~w = ( i, , 1 + i ) (a) Is S an orthogonal set? (4 points) S is an orthogonal set if and only if ~u ~v = ~u ~w = ~v ~w = 0. So, we take the dot product of each pair of vectors, remembering that for complex vectors, we must take the complex conjugate of the second vector: ~u ~v = u i v * i . ~u ~v = (1 i )(0) + (0)(1 i ) + ( i )(0) = ~u ~w = (1 i )( i ) + (0)(0) + ( i )(1 i ) = i + i 2 + i i 2 = ~v ~w = (0)( i ) + (1 + i )(0) + (0)(1 i ) = All the dot products are zero, thus S is an orthogonal set....
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This note was uploaded on 02/09/2010 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Feinberg,E
 Applied Mathematics

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