Linear Algebra- Final Exam Review Questions 2016

Linear Algebra- Final Exam Review Questions For the final exam, you may not use a calculator. The length of the exam is about an one and a half exams, and will be approximately weighted: 40% new material, then about 20% taken from the other three portions of the course (to add to 60%). 1. Show that Row(A) ⊥ Null(A). 2. 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. 3. Find the line of best fit for the data: x 0 1 2 3 y 1 1 2 2   −3 0 4. Let A = . (a) Is A orthogonally diagonalizable? If so, orthogonally diagonalize 0 0 it! (b) Find the SVD of A. 5. Let V be the vector space spanned by the functions on the interval [−1, 1].  1, t, t2 Use Gram-Schmidt to find an orthonormal basis, if we define the inner product: Z 1 2f (t)g(t) dt hf (t), g(t)i = −1 6. Let v1 , . . . , vp be orthonormal. If x = c1 v1 + c2 v2 + · · · + cp vp then show that kxk2 = |c1 |2 + · · · + |cp |2 . (Hint: Write the norm squared as the dot product). 7. Short answer: (a) If ku + vk2 = kuk2 + kvk2 , then u, v are orthogonal. (b) 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? (c) Explain why the image of a linear transformation T : V → W  1 2 (d) Is the following matrix diagonalizable? Explain. A =  0 5 0 0 is a subspace of W  3 8  13 (e) 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 (f) i. Suppose matrix Q has orthonormal columns. Must QT Q = I? ii. True or False: If Q is m × n with m > n, then QQT = I. iii. Suppose Q is an orthogonal matrix. Prove that det(Q) = ±1.   1 1 2 2 8. Find a basis for the null space, row space and column space of A, if A =  2 2 5 5  0 0 3 3 9. 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   1       x1 =   1  , x2 =  1  , x3 =  1 2 Gram-Schmidt (you might  1 1  , 1  1 10. 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 ? 11. 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) 12. 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) 13. Let v be a vector in IRn so that kvk = 1, and let Q = I − 2vvT . Show (by direct computation) that Q2 = I. 14. 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. 15. If B has linearly dependent columns, show that AB has linearly dependent columns. Hint: Consider the null space. 16. If λ is an eigenvalue of A, then show that it is an eigenvalue of AT . 2     2 −2 17. 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 18. 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). 19. Let 1, t be two vectors in C[−1, 1]. Find the length between the two vectors and the cosine of the angle between them using the standard inner product (the integral). Find the orthogonal projection of t2 onto the set spanned by {1, t}. 20. Define an isomorphism. Why is an isomorphism important? 21. Let  B= 1 −3   , 2 −8    −3 , 7 Find at least two B−coordinate vectors for x = [1, 1]T . 22. Let U, V be orthogonal matrices. Show that U V is an orthogonal matrix. 23. In terms of the four fundamental subspaces for a matrix A, what does it mean to say that: • Ax = b has exactly one solution. • Ax = b has no solution. • In the previous case, what is the “least squares” solution? What quantity is being minimized? • Ax = b has an infinite number of solutions. 24. Let T be a one-to-one linear transformation for a vector space V into IRn . Show that for u, v in V , the formula: hu, vi = T (u) · T (v) defines an inner product on V . 25. Describe all least squares solutions to x+y =2 x+y =4 26. Let u = [5, −6, 7]T . Let W be the set of all vectors orthogonal to u. (i) Geometrically, what is W ? (ii) Find the projection of x = [1, 2, 3]T onto W . (iii) Find the distance from the vector x = [1, 2, 3]T to the subspace W . 27. Can the SVD be used to determine if a matrix A is invertible? How? 3