Set 7 Solutions.pdf - ESE 318-02 Fall 2018 Homework...

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ESE 318-02, Fall 2018 Homework Set #7 (10 problems) Due Wednesday by noon, Oct. 17 1 . (a) Zill 7.4.6. (b) Zill 7.4.13. 2 . (a) Zill 7.4.21. (b) Zill 7.4.22. 3 . Zill 7.4.50. 4 . (a) Zill 7.5.4. (b) Zill 7.5.12. (c) Zill 7.5.15. (d) Zill 7.5.16. 5 . (a) Zill 7.5.44. (b) Zill 7.5.46. 6 . Zill 7.5.57. 7 . Consider these two lines that go through a common point. 3 , 2 , 1 5 , 2 , 1 : 2 Line 4 5 1 2 3 1 : 1 Line t z y x (a) Write an equation of a line that is perpendicular to both Line 1 and Line 2 and goes through that same common point. (b) Write an equation of a plane that contains Line 1 and Line 2. (c) Write an equation of a plane parallel to the plane in part (b), but that goes through the point (1,2,3). 8 . (a) Zill 7.6.24(b). (b) Zill 7.6.25. (c) Zill 7.6.26. 9 . For each of the following sets, V , determine if the given set is a Vector Space (Subspace) and show your work. If it is a Vector Space, find its dimension and write a basis set. (Each is a subset of a Vector Space, so you only need to check the closure axioms. Also, assume the standard operations of addition and multiplication.) (a) All vectors, 3 2 1 , , v v v v , in R 3 , such that 5 v 1 3 v 2 + 2 v 3 = 0. (b) All 2 X 3 matrices with all non-negative elements. (c) All upper triangular 2 X 2 matrices. (d) All skew-symmetric 2 X 2 matrices. (e) All functions   x b x a x f sin cos with any constants a and b . 10 . Let V be the set of all 2 X 2 matrices in which the elements of each row sum to 0. (a) Show that V is closed under scalar multiplication. (b) Show that V is closed under addition. (c) Find the dimension of V . (d) Find a basis for V . (e) Express 7 7 3 3 as a linear combination of your basis from (d).
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ESE 318-02, Fall 2018 Solutions 1 . (a) Zill 7.4.6. 10 , 6 , 14 10 6 14 2 12 10 4 15 1 1 3 2 5 1 4 k j i k j i k j i b a (b) Zill 7.4.13. The cross product (and any scalar multiple) is perpendicular to both. 5 , 2 , 3 Also, 5 , 2 , 3 5 2 3 7 2 4 2 4 7 1 1 1 4 7 2 k j i k j i k j i b a 2 . (a) Zill 7.4.21. 0 , 2 , 1 2 2 2 2 j i i j j k i k j i k (b) Zill 7.4.22. 0 , 0 , 0 0 i i k j i 3 . Zill 7.4.50. Using P 2 as the originating point.   185 16 25 144 4 6 1 12 1 4 2 3 0 1 1 , 4 , 2 6 , 0 , 0 5 , 4 , 2 3 , 0 , 1 6 , 0 , 0 3 , 0 , 1 2 1 2 1 2 1 2 1 2 1 k j i k j i b a b a A
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ESE 318-02, Fall 2018 4 . (a) Zill 7.5.4. Multiple possibilities. Two listed here.
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