Linear Algebra- Final Exam Review Questions 2011

Linear Algebra- Final Exam Review Questions For the final exam, you may use a calculator- You may not use a calculator that performs symbolic computations (like the derivative/integral). Your calculator may be a graphing calculator (no graphing required, however). 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 first for the data: x 0 1 2 3 y 1 1 2 2 4. 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). 5. 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? (f) Let Q be a matrix with 3 non-zero orthonormal columns. i. Can Q have fewer than 2 rows? Why or why not? ii. True or False: If Q is m × n with m > n, then QQT = I.   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 1 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   1       x1 =   1  , x2 =  1  , x3 =  1 2 Gram-Schmidt (you might  1 1  , 1  1 8. 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 ? 9. 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) 10. 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). 11. Let v be a vector in IRn so that kvk = 1, and let Q = I − 2vvT . Show (by direct computation) that Q2 = I. 12. 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. 13. If B has linearly dependent columns, show that AB has linearly dependent columns. Hint: Consider the null space. 14. If λ is an eigenvalue of A, then show that it is an eigenvalue of AT .     2 −2 15. 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. 2       a b c a + 2g b + 2h c + 2i g h i 16. 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). 17. 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}. 18. Define an isomorphism: 19. Let  B= 1 −3   , 2 −8    −3 , 7 Find at least two B−coordinate vectors for x = [1, 1]T . 20. Let U, V be orthogonal matrices. Show that U V is an orthogonal matrix. 21. 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 b= 0  c= 0  a= 3  −1 1 1 22. 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 . 23. Describe all least squares solutions to x+y =2 x+y =4 24. 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 . 3