Linear Algebra- Final Exam Review Questions 2010

Linear Algebra- Final Exam Review New material since Exam 3: 1. Definitions: Gram-Schmidt, QR, least squares solution, normal equations, 2. The Spectral Theorem is the central theorem of 7.1, and the SVD is the central theorem of 7.4 (know what the factorization produces), and the connection of the SVD to the 4 fundamental subspaces. For earlier material, look over the concept reviews and review questions from the previous exams. You should also be sure and look over your old exams and homework. Some “random” questions: 1. Let A be invertible. Show that, if v1 , v2 , v3 are linearly independent vectors, so are Av1 , Av2 , Av3 . NOTE: It should be clear from your answer that you know the definition. 2. Find the line of best first for the data: x 0 1 2 3 y 1 1 2 2   −3 0 3. Let A = . (a) Is A orthogonally diagonalizable? If so, orthogonally diagonalize 0 0 it! (b) Find the SVD of A. 4. Let V be the vector space spanned by the functions: f1 (x) = x sin(x) f2 (x) = x cos(x) f3 (x) = sin(x) f4 (x) = cos(x) Define the operator D : V → D as the derivative. (a) Find the matrix A of the operator D relative to the basis f1 , f2 , f3 , f4 (b) Find the eigenvalues of A. (c) Is the matrix A diagonalizable? 5. Short answer: (a) Let H be the subset of vectors in IR3 consisting of those vectors whose first element is the sum of the second and third elements. Is H a subspace? (b) Explain why the image of a linear transformation T : V → W  1 2 (c) Is the following matrix diagonalizable? Explain. A =  0 5 0 0 1 is a subspace of W  3 8  13 (d) If the column space of an 8 × 4 matrix A is 3 dimensional, give the dimensions of the other three fundamental subspaces. Given these numbers, is it possible that the mapping x → Ax is one to one? onto?   1 1 2 2 6. Find a basis for the null space, row space and column space of A, if A =  2 2 5 5  0 0 3 3 7. Find an orthonormal basis for W = Span {x1 , x2 , x3 } using wait until the very end to normalize all vectors at once):      0 0  0      , x2 =  1  , x3 =  x1 =   1   1   1 2 8. Find the QR factorization of the matrix A[x1 x2 Gram-Schmidt (you might  1 1  , 1  1 x3 ], defined in the previous problem. 9. Given the QR factorization of a matrix A, show that the least squares solution to Ax = b is x̂ = R−1 QT b. (Hint: Start with Ax̂, and see where you go). 10. Using the QR factorization in the previous problem for matrix A, find the least squares solution to Ax = b, where b = [1, 0, 0, 0]T . 11. Let Pn be the vector space of polynomials of degree n or less. Let W1 be the subset of Pn consisting of p(t) so that p(0)p(1) = 0. Let W2 be the subset of Pn consisting of p(t) so that p(2) = 0. Which if the two is a subspace of Pn ? 12. For each of the following matrices, find the characteristic equation, a basis for each eigenspace:      1 0 7 2 3 −1 A= B= C= 0 2 −4 1 1 3 1 0 the eigenvalues and  1 0  1   p(−1) 13. Define T : P2 → IR3 by: T (p) =  p(0)  p(1) (a) Find the image under T of p(t) = 5 + 3t. (b) Show that T is a linear transformation. (c) Find the kernel of T . Does your answer imply that T is 1 − 1? Onto? (Review the meaning of these words: kernel, one-to-one, onto) (d) Find the matrix for T relative to the basis {1, t, t2 } for P2 . (This means that the matrix will act on the coordinates of p). 2 14. Let v be a vector in IRn so that kvk = 1, and let Q = I − 2vvT . Show (by direct computation) that Q2 = I. 15. Let A be m × n and suppose there is a matrix C so that AC = Im . Show that the equation Ax = b is consistent for every b. Hint: Consider ACb. 16. If B has linearly dependent columns, show that AB has linearly dependent columns. Hint: Consider the null space. 17. If λ is an eigenvalue of A, then show that it is an eigenvalue of AT .     2 −2 18. Let u = and v = , Let S be the parallelogram with vertices at 0, u, v, and 1 1 u + v. Compute the area of S.       a b c a + 2g b + 2h c + 2i g h i 19. Let A =  d e f , B =  d + 3g e + 3h f + 3i , and C =  2d 2e 2f . g h i g h i a b c If det(A) = 5, find det(B), det(C), det(BC).        3 4 4 6 20. Let B = , and C = . Write down the matrices that −5 −6 5 7 take [x]C to [x]B and from [x]B to [x]C . 21. Define an isomorphism: 22. Let  B= 1 −3   , 2 −8    −3 , 7 Find at least two B−coordinate vectors for x = [1, 1]T . 23. Let B = {b1 , . . . , bn } be a basis for vector space V . Explain why the B−coordinate vectors of {b1 , . . . , bn } are the columns of the n × n identity matrix: 24. Find the volume of the parallelepiped formed by 0, a,b, c, a + b, c + b, c + a, and the sum of all three.       0 −2 −1 a= 3  b= 0  c= 0  −1 1 1 3