Math 19B: linear algebra and probability Oliver Knill, Spring 20115/6/2011: Final examYour Nam e:• Please enter your name in the above box. You have to do this exam on your own.• You can write the solutions on your own paper. Make sure you write your name on eachpage and staple it together at the end.• Please write neatly Answers which are illegible for the grader can not be given credit.• Unless stated, give computations, details reasons. The suggested exam time is Friday, the6th of May.• No calculators, computers, or other electronic aids are allowed. There is no time limit but3 hours should suffice. Bring your exam to my office in 434 on Monday May 9th, 2011 untilnoon 12. I need to get the paper in Person this can be done on Friday at 4 or Mondaybetween 10 and 12 AM.• The exam is op en book in the sense that you can use any of the distributed class lecturenotes and any of your handwritten notes but no other books or text resources.1 202 103 104 105 106 107 108 109 1010 1011 1012 1013 10Total: 140Problem 1) TF questions (20 points)1)TFA random variable on the probability space Ω = {1, 2, . . . , 10 } satisfyingP[X = k ] = 10!/(k!(10 − k)!210) has the expectation E[X] = 10/2.2)TFThere is a system of linear equations A~x =~b which has exactly 2 solutions.3)TFIf the sum of all geometric multiplicities o f a 3 × 3 matrix A is equal to 3,then A is diagonalizable.4)TFIf a system of linear equations Ax = b has two different solutions x, thenthe nullety of A is positive.5)TFIf a matrix A is invertible then its row r educed echelon form is also invert-ible.6)TFAll symmetric matrices are invertible.7)TFAll symmetric matrices have simple spectrum in the sense that all of theeigenvalues are different.8)TFThe geometric multiplicity is always la r ger or equal than the algebraic mul-tiplicity.9)TFThe Google matrix is a Markov matrix10)TFIf a 3 ×3 matrix has positive entries and is symmetric, then all eigenvaluesare different.11)TFIf a 2 × 2 mat rix has positive entries, then both eigenvalues are real anddifferent.12)TFIf two random variables X, Y are independent then σ[X+Y ] = σ[X]+σ[Y ].13)TFAn exponentially distributed random variable has positive expectation.14)TFThe variance of the Cauchy distributed random variable is 1.15)TFIf two events A, B satisfy that P[A|B] = P[A] then A, B are independent.16)TFFor the Monty-Hall problem, it does not matter whether you switch or not.The winning chance is the same in both cases.17)TFA random variable on a finite proba bility space {1, 2, 3 } can be interpretedas a vector with three compo nents.18)TFThe composition of two r eflections in the plane is an orthogonal transfor-mation.19)TFIf P[A|B] and P[A] and P[B] are known, then we can compute P[B|A].20)TFIf the P -value of an experiment is less than 15 percent, then the null hy-pothesis is rejected.21)TFIts not 15 percent.Problem 2) (10 points)a) (5 points) Which ma t r ices are diagonalizable? Which ones are Markov matrices?Matrix diagonalizable Markov"1/2 2/31/2 1/3#"1/2 2/31/2 1/2#"1/2 −2/31/2 −1/3#"1 20 1#"1 00 1#b) (3 po ints) Match the transformation with the trace and determinant:trace det enter A-F1 00 -10 11 1/42 1-2 1label transformationA reflection at the originB rotation by -90 degreesC projection onto x-axesD dilation by 1/2E reflection at a lineF vertical shearc) (2 points) Draw the image of the picture to the left under the linear transformation given bythe matrix A ="1 1−1 2#. Make sure to indicate clearly which arrow goes where.Problem 3) (10 points) Systems of linear equationsConsider the following system of linear equations:3x + y + z = λxx + 3y + z = λyx + y + 3z = λzYou can interpret the above system as a n eigenvalue problem Ax = λx for a 3 ×3 matrix A. Whatare the eigenvalues and what are the eigenvectors?Problem 4) (10 points) Bayesan statisticsWe throw 5 fair dice. Let A be the event that the sum of the first four dice is 5. Let B be theevent that the sum of the last two dice is 6.a) (4 points) Find the probabilities P[A], P[B].b) (3 po ints) Find the conditional probability P[B|A].c) (3 points) What is the conditional probability P[A|B]?Problem 5) (10 points) Basis and ImageDefine the solar matrixA =2 1 1 1 1 1 1 1 1 11 1 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 01 0 0 0 1 0 0 0 0 01 0 0 0 0 1 0 0 0 01 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 01 0 0 0 0 0 0 0 1 01 0 0 0 0 0 0 0 0 1.a) (4 po ints) Find a basis for the eigenspace to the eigenvalue 1 that is the kernel of B = A −I10.b) (3 po ints) Find a basis for the image of B.c) (3 points) Find the determinant of A.Problem 6) (10 points) Inverse and Coordinatesa) (5 points) Verify that ifA =B 0 00 C 00 0 Dwhere B, C, D are 2 × 2 matrices, thenA−1=B−10 00 C−100 0 D−1.b) (5 po ints) Invert the matrixA =2 1 0 0 0 03 2 0 0 0 00 0 3 2 0 00 0 0 3 0 00 0 0 0 3 −10 0 0 0 1 3.Problem 7) (10 points) Expectation, Va r iance, CovarianceWe throw 5 fair dice and call Xkthe number of eyes of dice k. As usual, the Xkare assumed to beindependent. Let X = X1+ X2+ X3denote the sum of the first 3 dice and Y = X4+ X5denotethe sum of the last two dice. Find the correlation Cov[X, Y ] between the two random variablesX, Y .Problem 8) (10 points) Eigenvalues and Eigenvectorsa) (7 points) Find all the eigenvalues and eigenvectors of the matrixA =0 3 0 0 0 0 44 0 3 0 0 0 00 4 0 3 0 0 00 0 4 0 3 0 00 0 0 4 0 3 00 0 0 0 4 0 33 0 0 0 0 4 0.Hint: Write A as 3Q + 4QTfor some orthogonal matrix Q for which you know how to computethe eigenvalues and eigenvectors.b) (3 po ints) Find the determinant of A.Problem 9) (10 points) Determinantsa) (3 points) Find the determinant of the matrixA =0 6 0 00 0 7 00 0 0 93 0 0 0.b) (3 po ints) Find the determinant of the matrixA =9 1 1 1 1 1 1 11 9 1 1 1 1 1 11 1 9 1 1 1 1 11 1 1 9 1 1 1 11 1 1 1 9 1 1 11 1 1 1 1 9 1 11 1 1 1 1 1 9 11 1 1 1 1 1 1 9.c) (4 points) Find the determinant of the matr ixA =1 2 3 4 5 6 7 8−1 0 3 4 5 6 7 8−1 −2 0 4 5 6 7 8−1 −2 −3 0 5 6 7 8−1 −2 −3 −4 0 6 7 8−1 …