MATH 218 Exam 2 (common), Spring 2011

Math 218 - Quiz 2 (Spring 2011) Common Exam - Version A i- You must show your work to receive credit. Correct answers with inconsistent work will receive no credit. ii- This is a closed book exam and no calculators are allowed. iii- Please turn off your cell phone. iv- The exam is out of 50. v- There are 5 questions in total. All questions have several parts to them. Make sure you attempt them all. vi- Write you section number (see the table below) on the coloured booklet containing your answers. vii DO NOT LOOK AT THE QUESTIONS UNTIL TOLD TO DO so Time Instructor Section Number TR , 12:30 — 13:45 Rana Nassif ZF TR, 11:00 - 12:15 Rana Nassif ) MWF , 10:00 - 10:50} Sara Abu Diab 3 MWF , 11:00 - 11:50] Monique Azar 4 MWF , 08:00 — 08:50|Friedmann Brock 5 MWF , 13:00 — 13:50/Michella Bou Eid 6 MWF , 14:00 — 14:50|Michella Bou Eid 7 MWF , 09:00 — 09:50] + Faleh Taha 8 TR , 09:30 — 10:45 Rana Nassif 9 MWF , 12:00 — 12:50] Monique Azar 10 MWF , 14:00 - 14:50] Tamer Tlas 11 1. (5 points each) Find a basis and the dimension for each of the following two vector spaces: i- The span of the following three 3-vectors 1 1 3 a) Gees (aie! ’ w= 1-0 > we 2 0 1 1 ii- The span of the following three polynomials pi(z) =1+2 , p2(a) = x? ‘ p3(c) = 27 +2042 2. (5 points each) Consider the following: i- If U is a vector subspace of V and V is a vector subspace of W, does it follow that U is always a vector subspace of W? If yes provide a proof, if no supply a counterexample. ii- Suppose U and V are vector subspaces of R”. Assume that there is a vector in V which is not in U (i.e. assume that the set of all vectors which are in V but not in U is not empty). Does it follow that the set of all vectors in V which are not in U is a vector subspace of R”? Justify your answer. 3. (5 points each) Let A be a 4 by 3 matrix whose entries are not all zeros. pay 1 sk i- Is the set of all 3-vectors x satisfying Ax = a vector subspace of R°? Justify your 0 af answer. 0 a Seer 0 ii- Is the set of all 3-vectors x satisfying Az = a vector subspace of R°? Justify your 0 0 answer. 4. (3 points each) Answer the following questions: i- Suppose V is the vector space of all 7 by 8 matrices. Let F: V — R be the function that assigns to each matrix the number which is the sum of the entry in the top left corner with the entry in the bottom right corner. Is F a linear transformation? Justify your answer. ii- Let V be as before. Let G: V > R be the function that assigns to each matrix the number which is the product of the entry in the top left corner with the entry in the bottom right corner. Is G a linear transformation? Justify your answer. iii- Let U be the vector space of all polynomials. Let H : U > U be the function which is defined via the formula H(p(x)) = x p(x). Is H a linear transformation? Justify your answer. 5. Consider the following: i- (4 points) If U and V are vector spaces, is their intersection UNV a vector space? (The intersection is the set of all common elements). ii- (4 points) Assume you have a collection of linearly independent vectors U1,+++)Un,U1,+++,Um- Let V be span of v4,..., vn and let U be the span of uj,..., Um. Describe the intersection of U and V and find its dimension. iii- (3 points) Assume now that instead you are given that v1,...,Un,U1,-.-,Um-—1 is a linearly independent collection of vectors, and that ui,...,Um is also a linearly independent collection of vectors, and also that um is in the span of v1,...,Un. As before let V be the span of v},...,Un and U be the span of w1,...,Um. Describe the intersection of U and V, and find its dimension.