Final Exam - Take Home Portion Math 2, Summer 207 Name: Directions: Complete a total of 5 problems. Problem must be completed. The remaining problems are categorized in four groups. Select one problem from each of the four groups. Show all your work. You may use your notes and textbook, as well as previous problem sets and the midterm. No other resources are allowed. You must work independently. P. Let 2 3 0 A = 6 7 0. 6 5 2 a) Determine the nullity(a) and a basis for ker(a). b) Determine rank(a) and a basis for im(a). c) Determine if A is invertible. d) Compute the eigenvalues and eigenvectors of A. e) If λ,λ 2,λ 3 are the eigenvalues of A with eigenvectors v,v 2,v 3, respectively, show (or clearly state how you know) that {v,v 2,v 3 } is a basis for R 3.
Group P 2. Let T : P 2 P 2 be the linear transformation defined by T (f ) = (a + b + c)t 2 + (a 2b)t + 2c, where f (t) = at 2 + bt + c. Let B = {,t,t 2 } be the standard basis of P 2. a) Determine the standard matrix representation of T. b) Denoting the standard matrix representation of T by A, determine the eigenvalues and eigenvectors of A. You should find that A has two distinct eigenvalues; one positive, one negative. Let λ < 0 be the negative eigenvalue. c) Let v be an eigenvector corresponding to the negative eigenvalue λ. Let f P 2 be the polynomial with coordinate vector v, so [f ] B = v. Show that T (f ) = λ f. P 3. Let T : P 2 P 2 be defined by T (f ) = f (t) 2f (t) 3f (t). a) Determine the standard matrix representation of T. b) Let A be the standard matrix representation of T. Determine the eigenvalues and eigenvectors of A. c) Is the matrix A diagonalizable? If so, diagonalize A by defining an appropriate matrix P and show that P AP = D, where D is a diagonal matrix.
Group 2 P. Let v R 2 be a fixed vector. For any u R 2 define the orthogonal projection of u onto v by proj v u = u,v v,v v, where u,v = u v = u v + u 2 v 2 is the standard inner product (or dot product) on R 2. a) Take and compute proj v u. u = 3 3, v = b) For the vector u = compute the orthogonal projection of u onto the x-axis by computing proj e u, where e = is the first standard basis vector for R 2. Graph the vectors u and proj e u. 3 0 c) Compute the orthogonal projection of u onto the y-axis by computing proj e2 u. Graph the vector proj e2 u on your sketch from part b. d) Show that, for a fixed vector v R 2, proj v : R 2 R 2 is a linear transformation. e) For an arbitrary (fixed) vector v = ( v determine the standard matrix representation of proj v. v 2 ) f) Let P be the standard matrix representation of proj v. Show that P 2 = P. g) Recall that two vectors x,y R n are orthogonal if x,y = 0. Show that for any u,v R 2 the vectors u proj v u and v are orthogonal.
P 5. Let u,v R n be two vectors. The angle ϕ between u and v satisfies cosϕ = u,v u v, (0.) where u = u,u = u 2 + u2 2 + + u2 n is the norm of u. For part (a), let u = 2 2 =. 2 a) Compute the angle ϕ between u and u 2. Sketch the vectors. For parts (b)-(c), let 3 6 v = 2, v 2 =, v 3 = 2. 2 b) Compute the angle ϕ between v and v 2. c) Compute the angle ϕ between v and v 3. d) Referring to Problem where u = ϕ = π sing equation (0.)., verify that the angle between u 2 and [u ] B is
Group 3 P 6. Determine the eigenvalues of the matrix 0 0 0 J = 0 0. 0 0 Then, show that J is diagonalizable, by constructing a matrix P whose columns are eigenvectors of J and computing P JP. P 7. Let 2 2 A = 0 2. 0 0 a) Determine the eigenvalues of A and the corresponding eigenvectors. b) Is the matrix A diagonalizable? If so, diagonalize A by defining an appropriate matrix P.
Group P 8. For a fixed vector v R n, let S v = {u R n u,v = 0} be the set of all vectors which are orthogonal to v. Show that S v is a subspace of R n. P 9. Let cosϕ sinϕ A =. sinϕ cosϕ a) Recall that this particular matrix A represents rotation by the angle ϕ R. Notice that A is invertible for any ϕ R (if you are not sure, check that A is invertible). By the invertible matrix theorem, the columns of A form a basis of R 2, no matter what value of ϕ R we choose. Take ϕ = π and let v,v 2 be the first and second columns of A, respectively. b) Let u = 2 2 = 3 = 2 =. Considered as points in R 2, draw the rectangle with vertices u,u 2,u 3,u. c) Let B = {v,v 2 } be the basis corresponding to the columns of A and let S = {e,e 2 } be the standard basis of R 2. Determine the coordinates of the vectors u,u 2,u 3,u with respect to the basis B. Considered as points in R 2, draw the rectangle with vertices [u ] B,[u 2 ] B,[u 3 ] B,[u ] B on the same sketch from part (b).