1 1. (6+4+4+6) Let P3 denote the vector space of all third degree or less polynomials. (a) Find two linearly independent vectors in P3 that interpolate the data points (1, 2), (2, 1) and (3, 2). (b) Is p = 1 x + x2 x3 in span(p1 , p2 , p3 ), where p1 = 9 4x 3x2 2x3 , p2 = 1 x3 and p3 = 2 + x + x2 ? Explain (c) Show that the subset S of P3 consisting of all polynomials p(x) with the property that p(1) = 0 is a subspace of P3 . (d) Find a basis for S. Explain why it is a basis. What is the dimension of S? 2. (8+2) Let A = 1 2 4 3 4 10 1 1 3 2 3 7 (a) Find a basis for each of the four fundamental subspaces of A. (b) Find a basis for the row space of A that is a subset of the rows of A. 3. (2+4+3+6) Consider the 8 8 system of equations AX = B, where A= 100 0 0 0 0 0 0 0 1 101 0 0 0 0 0 0 1 1 102 0 0 0 0 0 1 1 1 103 0 0 0 0 1 1 1 1 104 0 0 0 1 1 1 1 1 105 0 0 1 1 1 1 1 1 106 0 1 1 1 1 1 1 1 107 , B= 8.0 6.1 5.01 4.001 3.0001 2.00001 1.000001 0.0000001 (a) Why does this system have a unique solution? (b) Find the unique solution to this system of equations. How did you do this? (c) If r denotes the residual, find ||r||1 , ||r||2 and ||r|| . (d) How good of an approximation is your solution in (3b) to the actual solution. Explain. 2 1 1 1 4. (4+4+4+6) Let C = 1 3 12 2 1 1 8 2 (a) Show that X = 3 5 1 0 1 1 0 1 is a basis for N (C). 2 (b) Show that Y = (c) Let v = 4 10 3 5 1 3 1 2 0 2 1 3 is also a basis for N (C). N (C). Find the coordinates of v in the basis X and the basis Y . (d) Find the change of coordinate matrices from X to Y and from Y to X. 5. (3+4+4) Consider the matrix 1 Q= 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (a) Show that Q is an orthogonal matrix. (b) Let B be the basis of R4 consisting of the columns of Q. Let v R4 be the vector whose coordinates in the standard basis are (2, 4, 6, 2)T . What are the coordinates of v in the basis B? 1 2 2 0 1 0 1 1 0 . Find the reduced QR-factorization of W , i.e., find a 6 3 matrix Q 6. (4) Let W = 1 3 2 0 1 2 1 2 0 that has orthonormal columns and an invertible, upper-triangular matrix R such that W = QR. 7. (4+4) Let A be an n n symmetric matrix with positive eigenvalues. Recall that the standard inner P product of two vectors x, y in Rn is defined as x y = xT y = n1 xi yi . (a) Show that Ax x > 0 for every nonzero x Rn . 4 2 7 3 2 8. (4) Let F = 3 3 7 . Find a matrix B (which has real entries) such that B = F . 12 2 15 9. Find a 2 2 matrix A (having no zero entries) such that A2 + A 6I = 0, where I is the 2 2 identity matrix. (Hint: Suppose A is diagonal first) 3 10. (4+8) Let A be the 4 4 matrix as below that is missing the bottom two rows 1 -2 A= 4 -8 -2 5 1 1 (a) Fill in the bottom two rows above with non-zero rows so that the rank of A is 2. Justify your answer. (b) Find a basis for the four fundamental subspaces of A. be the n 1 vector, X = (1, 1, 1, . . . , 1)T 11. (4+2+2+4) Let H be the n n Hilbert matrix, let X Therefore X is the unique solution to the system of equations and let B be the vector B = H X. HX = B. Now use your favorite method in Matlab to solve the linear system of equations HX = B. Naturally we do expect there to be round-off error, so the solution we find may not be exactly equal = (1, 1, 1, . . . , 1)T . to X is the true solution and r (a) Fill in the table below. Recall that X is the solution from Matlab, X is the residual. 2 n ||r||2 ||X X|| 3 6 9 12 (b) What conclusion(s) do you draw from this? (c) Let B be the vector in R9 that contains the digits of your student number. Solve the 9 9 system HX = B. (d) Discuss how accurate your solution to the above problem is. 12. (4+4+4+4) Let P be the unique polynomial of degree two that interpolates the data points (1, 4), (2, 3), (3, 0) (a) What are the coordinates of P in the basis B 1 = {1, x, x2 }? (b) What are the coordinates of P in the basis B 2 = {1, x 1, (x 1)(x 2)}? 4 (c) What are the coordinates of P in the basis B 3 = {1, x 2, (x 2)(x 3)}? (d) Find the change of basis matrix from B 2 to B 3 coordinates. 2 1 1 13. (2+4+2+4+4) Let A = 0 1 1 0 0 1 (a) What are the eigenvalues of A? (b) Find a basis for the eigenspaces, N (A I), for each eigenvalue, , of A. (c) Is A diagonalizable? Explain. (d) Is AT A diagonalizable? Explain. (e) Find a 3 3 matrix B having the same eigenvalues as A such that B is not diagonal but is diagonalizable. Explain. x1 3 14. (4+4+2+4+4) Let S be the subset of all vectors, X = x2 , in R such that x3 = 2x1 x2 . x3 (a) Show that S is a subspace of R3 . (b) Find a basis for S. (c) What is the dimension of S? (d) Find an orthonormal basis for S. (e) Find the change of coordinates matrix from your basis in (14b) to your basis in (14d). 15. (4+4+4+4) Consider the two linearly independent vectors v1 = 3 1 0 2 , v2 = 1 0 1 4 (a) Find a 4 4 matrix A1 such that the column space of A1 = span{v1 , v2 }. Explain. (b) Find a 4 4 matrix A2 such that the row space of A2 = span{v1 , v2 }. Explain. (c) Find a 4 4 matrix A3 such that the null space of A3 = span{v1 , v2 }. Explain. (d) Find a 4 4 matrix A4 such that the left null space of A4 = span{v1 , v2 }. Explain. 16. (8) Create a Matlab function with prototype [X r] = orthoBasisRow(A). Here the input, A, is a given m n matrix, r is the rank of A and X is a matrix whose columns form an orthonormal basis of A. Hand in your file orthoBasisRow.m to the Dropbox folder entitled Final Exam. Your code should NOT display any intermediate results. Marks will be deducted if this happens. 5 Part B: Multiple Choice and True/False 1. If u and v are orthogonal vectors in an inner product space then ||u + v||2 = ||u||2 + ||v||2 . 2. If rank(A) = rank(AT ) then A must be a square matrix. 3. A set containing a single vector is linearly independent. 4. If a square matrix A is diagonalizable then AT is diagonalizable. 5. If E is an n n elementary martix and A is an m n matrix, then the row space of AE is the same as the row space of A. 6. The eigenvalues of a matrix A are the same as the eigenvalues of the reduced row echelon form of A. 7. If X1 and X2 are solutions to the system of equations AX = B then X1 X2 is a solution to the corresponding homogeneous system AX = 0. 8. If A2 is a symmetric matrix then A is a symmetric matrix. 9. The set of upper triangular n n matrices is a subspace of the vector space of all n n matrices. 10. If a linear system has more unknowns than equations then it never has a unique solution. For the following multiple choice questions assume that A is an m n matrix with rank(A) = r. 11. Suppose AX = B is consistent for every B in Rm . (a) rank(A) = m (b) rank(A) = n (c) R(A) = Rm (d) (11a) and (11c) (e) (11b) and (11c) 12. Suppose AX = B has at most one solution for every B in Rm . (a) rank(A) = m (b) rank(A) = n (c) R(A) = Rm (d) (12a) and (12c) (e) (12b) and (12c) 6 13. Suppose AX = B has a unique solution for every B in Rm . (a) rank(A) = m (b) rank(A) = n (c) R(A) = Rm (d) N (A) = {0} (e) All of the above 14. dim(R(A)) = (a) m (b) n (c) m r (d) n r (e) None of the above 15. dim(R(AT )) = (a) m (b) n (c) m r (d) n r (e) None of the above 16. dim(N (A)) = (a) m (b) n (c) m r (d) n r (e) None of the above 17. dim(N (AT )) = (a) m (b) n (c) m r (d) n r (e) None of the above