MATH 218 Final, Spring 2009

MATHEMATICS 218 SPRING SEMESTER 2008-09 FINAL EXAMINATION Time: 120 minutes Date: June 3, 2009 Name: ID Number: Circle your section number in the table below: Answer the following nine sets of questions on the allocated pages; the back of pages may be used if needed Instructors | Ejeili | Fuleihan | Itani | Karam | El Khoury | Lyzzaik | Nassif | Tannous Section 7 3 11 1 10 6 2 4 Section 5 12 8 9 [ QUESTION GRADE 1 /25 2 /20 3 /20 4 /20 5 /20 6 /20 7 /20 8 /25 9 /30 TOTAL GRADE /200 1. Let T : R* > R® the transformation defined by T(z, y,2z,w) =(a@+y—z2+u,2e+yt+4z+u,3r + y4 92). (a) Show that T is a linear transformation. (4 points) (b) Find the standard matrix [T]. (5 points) (c) Find bases for the kernel of and range of T. (12 points) (d) Find the rank and nullity of T. (4 points) 2. Let T': P) > R® be the function defined by the formula p(1) T(p(x)) = | p(2) |; p(3) here P, is the vector space of all real polynomials of degree at most 2. (a) Show that T is a linear transformation. (8 points) (b) Show that T’ is one-to-one. (8 points) (c) Show that T is onto. (4 points) 3. Let P3 be the set of all polynomials of degree at most 3, and let W = {ax* + ba? + ex +-d:b+c+d=0}. (a) Show that W is a subspace of P3. ( (b) Find a basis S for W. ( (c) Give one vector in P; but not in W. (2 points) (d) Complete S to a basis for P3. (6 points) 6 points) 4. Let 135 7 A=|2 0 4 2 3.28 7 (a) Find a basis for the row space of A and a basis for its orthogonal complement. (16 points) (b) Find a subset of the column vectors of A that forms a basis for the column space of A. (4 points) 9 5. Let P: be the set of all polynomials of degree at most 2, and let p(x) = ao + a2 + dgx? and q(x) = bo + bx + box? be in P. Define on Py the operation <p,q>= Agbp + 2a1b; + 3agbo. (a) Show that <.,. > is an inner product on P». (10 points) (b) Show that (abo + 2a1b, + 3agb2)? < (a? + 2a? + 3a3)(b2 + 2b? + 303). (5 points) (c) Determine the cosine of the angle between the polynomials l+a—2? and 1—-a2+2". (5 points) 10