Final Examination -NYC-5 - Linear Algebra I December 8 th 7. (4 points) Let A = has: (a) a unique solution. a a (b) infinitely many solutions. (c) no solution. and b = 4. Find the value(s) of a for which the equation Ax = b. (4 points) Find the polynomial p(x) = x 4 + ax 3 + bx + cx + d such that p() = p( ) = p () = 7 and p ( ) = 3. 3. (6 points) Given the matrix A = (a) Col(A) (b) Row(A) (c) Nul(A T ) 4 8 4 4 3 find a basis for each of the following subspaces. 4. ( points) Suppose u is a solution to Ax = b and v is a solution to Ax =. Show that w = 3u 4v is a solution to Ax = 3b. 5. (6 points) Let T : R R denote the vertical expansion by a factor of and let S : R R denote the (counter-clockwise) rotation about the origin by π/ radians. (a) Find the standard matrix of the composite S T. (b) Let R denote the triangular region with vertices ( ) (4 ) and (3 ). Make two sketches one of R and the other of the image (S T )(R). [ B 6. Let A = C I where B is invertible. (a) (3 points) Find an expression for the partitioned matrix A. (b) (3 points) Use your work from part (a) to find the inverse of the matrix (c) ( point) Use your previous result to find the inverse of 4 5 3 4 5 3.. Page of 5
Final Examination -NYC-5 - Linear Algebra I December 8 th 7 7. Consider the matrix A = / 3 (a) (3 points) Find A by row reduction. (b) (4 points) Express the matrix A as a product of elementary matrices.. 8. (3 points) Let A and B be invertible n n matrices. Given that B is symmetric determine whether the matrix AB A T B is also symmetric. Justify your answer. 9. (3 points) Solve for the matrix X in the equation below: (3XB) + A = X Assume that all matrices involved are invertible. 3 3. (4 points) Given the matrix A = 3 4 3 find the determinant of A.. (6 points) Given A B and C are matrices such that det(a) = 3 det(b) = and det(c) = evaluate the following determinants. (a) det((3ab ) ) (b) det(ac + BC) (c) det(a +adj(a)). (7 points) Let H = {A M : det A = }. (a) Find two matrices in H neither of which is a scalar multiple of the other. (b) Is H closed under addition? (c) Is H closed under scalar multiplication? (d) Is H a subspace of M? 3. (4 points) Let v = [ 3 and S = {A M : Av = }. Find a basis for S. 4. (4 points) Let W be the set of all polynomials p in P 3 such that p() = and p ( ) =. Find a basis for W. Page of 5
Final Examination -NYC-5 - Linear Algebra I December 8 th 7 5. (6 points) Let u = 3 and v =. (a) Find an equation of the form ax + bx + cx 3 = d for the plane spanned by u and v. x 9 (b) Show that the line x = 6 + t is entirely contained on the plane spanned by u and v. x 3 4 3 6. Consider the plane P : x 4y + z = 6 the line L : x = 3 + t and the point Q( 6 ). (a) ( point) Find an equation for the line through the origin and the point Q. (b) ( points) Find the cosine of the angle between the plane P and the yz plane. (c) (3 points) Find the distance from the point Q to the line L. (d) (4 points) Find the point on the plane P that is closest to the point Q. (e) (3 points) Find an equation of the form ax + by + cz = d for the plane that contains the line L and is perpendicular to the plane P. 3 7. (4 points) Let u = and v = k. k (a) Find all values of k for which u and v are orthogonal. (b) Find a unit vector that is orthogonal to u. 8. (3 points) Let {u v w} be a set of linearly independent vectors in R 3. (a) Simplify u [(v u) (w u). (b) True or false: the parallelepiped with sides u v and w has the same volume as the parallelepiped with sides u v u and w u. 9. (3 points) Show that if {a b} is linearly independent then {a b a+b} is also linearly independent.. (4 points) Complete the following statements with must might or cannot as appropriate. (a) If A is a product of elementary matrices then det(a) (b) Two lines in R 3 that are both perpendicular to a third line equal zero. be parallel. (c) If matrix AB is invertible then A be invertible. (d) Given a linear transformation T : R 4 R the kernel of T be a plane. Page 3 of 5
Final Examination -NYC-5 - Linear Algebra I December 8 th 7 ANSWERS. (a) a a and a 4 (b) a = 4 (c) a = or a =. p(x) = x 4 x 3 3x + x + 3. (a) 4 8 3 4 (b) {[ [ } (c) 4. Aw = A(3u 4x) = 3Au 4Av = 3b 4 = 3b 5. (a) (b) [ [ = [ [ 6. (a) B CB I 7. (a) A = (b) A = (b) 4 6 4 / 5/6 /3 /6 /3 / / / 3/ 7/6 / 3/4 / / 3 4 (c) Transpose of answer in part (b) 4 Page 4 of 5
Final Examination -NYC-5 - Linear Algebra I December 8 th 7 8. Yes as (AB A T B) T = (AB A T ) T B T = A(B ) T A T B = A(B T ) A T B = AB A T B 9. X = A (I 3 B ). det(a) = 3. (a) [. (a) [ (b) No. [ [ 8 (many answers) H however (b) (c) 6 3 [ + [ = [ / H (c) Yes. Given A H k R det(ka) = k deta = k = (d) No as H is not closed under vector addition. 3. {[ 3 [ 3 } 4. {3x 3 + x 5x x 3 + 3x 5} 5. (a) x + x 5x 3 = (b) x + x 5x 3 = ( + 9t) + (6 + t) 5( + 4t) = 6. (a) x = t 6 (b) Using u = 4and v = cos(θ) = 5 6 ( (d) 3 4 3 7 ) (e) y + z = 7 3 7. (a) k = and k = 3 (b) / / (many answers) 8. (a) u (v w) (b) True (c) 3 units 9. Show that if c (a b) + c (a + b) = then c = and c =. c (a b) + c (a + b) = c a c b + c a + c b = (c + c )a + ( c + c )b = As a and b are linearly independent c +c = and c +c =. Solving this system of two equations gives c = and c =.. (a) cannot (b) might (c) might (d) might Total: points Page 5 of 5