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Math 205A Quiz 05 page 1 October 24, 2008 NAME
- Suppose V is a vector space and H is a subset of V. What are the three properties that H must satisfy in order to be a subspace of V? (List in the same order as we have in class)
(1)
- As in class, let F be the vector space whose members are functions f : R → R which are continuous. The figure below shows the graphs of two vectors u and v in F. On the empty grid to the right, show the graph of − 2 v. On the bottom grid, show the linear combination u + v. (Draw both answers with solid lines)
- Let H be the subset of all vectors in F whose graph is horizontal from x = 1 to x = 3. 3a. Is either of u or v from problem ( 2 ) in H? Explain both decisions.
Math 205A Quiz 05 page 2 October 24, 2008 NAME
3b. For each of the 3 properties in problem ( 1 ), decide if the property holds for H or does not. If it holds, explain why; if it does not, then give a counterexample (in pictures if you wish).
- Let S be the “script-S” vector space of all infinite sequences of real numbers we’ve discussed in class. Let H be the subset of S consisting of all sequences which change sign from term-to-term, that is, if s is a sequence in H, and some term of s is positive, the one after it is negative, and vice-versa. By default, we will also automatically add the sequence 0 = (0, 0 , 0 ,... ) to H.
4a. Is s 1 = (1, − 2 , 3 , − 4 , 5 , − 6 ,... ) in H?
4b. Is s 2 = (3, 0 , − 3. 1 , 0 , 3. 14 , 0 , − 3. 141 ,... ) in H?
For each of the 3 properties in problem ( 1 ), decide if the property holds for H or does not. If it holds, explain why; if it does not, then give a counterexample using specific sequences.
