UCLA MATH 131A - midterm1 solution (5 pages)

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midterm1 solution



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midterm1 solution

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Pages:
5
School:
University of California, Los Angeles
Course:
Math 131a - Analysis

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Linear Algebra Solution to Midterm 1 Problem 1 Let V a1 a2 a1 a2 R Define addition of elements of V coordinatewise and for a1 a2 in V and c R define n 0 0 if c 0 c a1 a2 ca1 ac2 if c6 0 Is V a vector space over R with these operations Justify your answer 5 points Solution No If c d R c d 6 0 c 6 0 d 6 0 then c d a1 a2 c d a1 a2 c d usually is not equal to c a1 a2 d a1 a2 ca1 da1 a1 a2 c d VS8 does not hold Problem 2 Let W1 and W2 be subspaces of a vector space V Prove that W1 W2 is a subspace of V if and only if W1 W2 or W2 W1 9 points Proof that W1 W2 or W2 W1 then W1 W2 W1 or W2 Since W1 and W2 are subspaces V we have W1 W2 is also a subspace of V Suppose that W1 W2 is a subspace of V Also suppose that W1 6 W2 and W2 6 W1 then there exist u v V such that u W1 W2 v W2 W1 u v W1 W2 u v W1 W2 If u v W1 then u u v W1 v W1 If u v W2 then u v v W2 u W2 Hence W1 W2 or W2 W1 Problem 3 Show that if S1 and S2 are arbitrary subsets of a vector space V then span S1 S2 span S1 span S2 9 points 1 P Pn Proof Let u span S1 S2 then u m i 1 ai vi j 1 bj wj for some scalars ai i 1 m bj j 1 n where vi i 1 m are in S1 and wj j 1 n are in P Pn S2 Since m a v is in span S and i i 1 i 1 j 1 bj wi is in span S2 we have u span S1 span S2 Hence span S1 S2 span S1 span S2 Now let v x y span S1 span S2 where x span S1 and y span S2 We Pm can write x i 1 ai vi for some scalars ai i 1 m and vi S1 i 1 m Pn and y j 1 bj wj for some scalars bj j 1 n and wi S2 j 1 n Then Pm Pn we can see that v x y a v i i i 1 j 1 bj wj is in span S1 S2 since vi i 1 m wj j 1 n are in S1 S2 Hence span S1 span S2 span S1 S2 Therefore span S1 S2 span S1 span S2 Problem 4 Prove that a set S is linear dependent if and only if S 0 or there exist distinct vectors v u1 u2 un in S such that v is a linear combination



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