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homework2 67 deloreda

# homework2 67 deloreda - x 1(1-i x 2 = 0 ——————...

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Math 67 Homework 1 Due January 23th IMPORTANT: Please hand-in calculational exercises separated from proof- writing exercises (there will be two piles). BONUS exercises (if any) don’t have to be handed-in. I separated the list of exercises in the three categories for your convenience. 1. Solve all calculational exercises 1(a), 2 in chapter 3 2. Solve calculational exercises 1,2,3(a,c,f),4(b,d,i,j) 5(c,h,i) 6(c,d), 7, 8(a,h,q). from chapter 12. 3. Solve the following system of linear equations using the factorization method. x 1 + x 2 + x 3 - 2 x 4 - 2 x 5 = 5 2 x 1 - x 2 - x 4 - 2 x 5 = 0 2 x 1 - x 3 - 2 x 4 - x 5 = 0 x 2 - x 3 - x 4 + x 5 = 0 x 1 - 3 x 2 +4 x 3 +6 x 4 = 0 - 2 x 1 +4 x 2 + x 3 +7 x 4 = 0 3 x 1 - x 2 +2 x 3 +5 x 4 = 0 - x 1 +2 x 2 +3 x 3 +7 x 4 = 0 4. For each of the following matrices find a RREF matrix equivalent to it and the associated factorization via a product of elementary matrices: A 1 = 1 2 1 0 - 1 0 3 5 1 - 2 1 1 A 2 = 2 0 i 1 - 3 - i i 1 1 5. Find all solutions of the system of equations: (1

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Unformatted text preview: x 1 + (1-i ) x 2 = 0 —————— 6. Solve proof-writing exercises 1,3 in chapter 3 7. Solve proof-writing exercises 1,3,4 from chapter 12. 1 8. Find two examples of 2 × 2 matrices with the property that A 2 = 0 but A 6 = 0. 9. For which values of k does the following system have a solution: x 1 + x 2 + x 3 + x 4 = 1 x 1 + x 3 = 1 x 2 + x 4 k 10. for which values of ( a, b, c ) does the following system have a solution? 3 x 1-x 2 +2 x 3 = a 2 x 1 + x 2 + x 3 = b x 1-3 x 2 = c 11. Suppose A is a 2 × 1 matrix and B is a 1 × 2 matrix. Show that AB is not invertible. 12. Recall that a square matrix is upper triangular if A ij = 0 for i > j . Show that a square upper triangular matrix is invertible if and only if each of its diagonal elements are non-zero. 2...
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homework2 67 deloreda - x 1(1-i x 2 = 0 ——————...

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