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Midterm 1-Fall 2007-Solutions

Midterm 1-Fall 2007-Solutions - AMS 510.01 Fall 2007...

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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 non-singular matrices, A and B , if A- 1 = B- 1 , then A = B • True. The matrix inverse is unique for any given (non-singular) 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 non-zero 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 non-zero complex number has a unique multiplicative inverse. • True. Just as the multiplicative inverse exists and is unique for all non-zero real numbers, the same is true of complex values. (g) The following matrix is non-singular: 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 n-square 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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Midterm 1-Fall 2007-Solutions - AMS 510.01 Fall 2007...

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