Math 19B: linear algebra and probability Oliver Knill, Spring 20113/23-25/2011: Midterm examYour Nam e:• Please enter your name in the above box. You have to do this exam on your own.• Answer each question on the same page as the question is asked. If needed, use the back orthe next empty page for work. If you need additional paper, write your name on it.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly Answers which are illegible for the grader can not be given credit.• Unless stated, give computations, details reasons.• No calculators, computers, or other electronic aids are allowed. There is no ho ur time limit.90 minutes should suffice a nd 3 hours are reasonable. You have to chose one day from the23-25th and do the exam during that day. Staple it together a nd bring it t o class on Friday.• The exam is open book in the sense that you can use the class lecture notes and anyhandwritten notes of yours but no other books or text resources.1 102 103 104 105 106 107 108 109 1010 10Total: 100Problem 1) TF questions (10 points)1)TFTwo events A, B which are disjoint satisfy P[A ∩ B] = P[A] + P[B].Explanation:2)TFTwo events A, B which are independent satisfy P[A ∩ B] = P[A] · P[B].Explanation:3)TFThe empty set is always an event in any probability space.Explanation:4)TFIf T is a linear map which satisfies T ("11#) ="1−1#and T ("1−1#) ="11#, then T is orthogona l.Explanation:5)TFThe vectors111,110,100are linearly independent.Explanation:6)TFThe expectation o f the ra ndo m variable X = (2, 3, 5) is 10/3.Explanation:7)TFThe standard deviation of the random variable X = (5, 6, 7) is 1/3.Explanation:8)TFLet A, B be arbitrary 2 × 2 matrices. If a vector x is in the kernel of A,then it is in the kernel of BA.Explanation:9)TFLet A, B be arbitrary 2 × 2 matrices. If a vector x is in the kernel of B,then it is in the kernel of BA.Explanation:10)TFThe least square solution of the linear system of equations Ax = y is a realunique solution of the system if A is invertible.Problem 2) (10 points)Match the matrices with their actions:A-J domain codomain A-J domain codomainA="1 02 1#B="1 01 2#C="1 01 1#D="0 11 0#E="0 11 2#F="0 12 1#G="1 11 0#H="1 10 1#I="1 11 2#J="1 12 1#Problem 3) (10 points) Systems of linear equationsa) (6 points) Find the general solution of the following system of linear equations using row re-duction.x + y + z + u + v = 4x − y + z − u + v = 0x − y + z = 2b) (2 points) The solutions ~x =xyzuvcan be written as ~x =~b +V , where~b is a particular solutionand V is a linear space. What is the dimension of V ?c) (2 points) Which of the three cases did appear: exactly one solution, no solution or infinitelymany solutions?Problem 4) (10 points) Random variables, independenceLets look at the following two vectors in R4:X =7−5−42, Y =1−12−2.a) (2 points) The two vectors span a linear space V . Write V it as an image of a matrix A.b) (3 points) Find the space W of all vectors perpendicular to X , Y . Write W as t he kernel of amatrix B.We can see these matrices also as random variables over the probability space {1, 2, 3, 4 }. Forexample X(1) = 7, X(2) = −5, X( 3) = −4, X(4) = 2.c) (2 points) Check that the two random variables X, Y are uncorrelated. What does Pythagorastell about Var[X + Y ]?d) (3 po ints) Are the two random variables X, Y independent random variables?Problem 5) (10 points) Basis and Imagea) (5 points) Find a basis for the kernel of the following diamond matrix:A =0 0 0 0 8 0 0 0 00 0 0 8 8 8 0 0 00 0 8 8 8 8 8 0 00 8 8 8 8 8 8 8 08 8 8 8 8 8 8 8 80 8 8 8 8 8 8 8 00 8 8 8 8 8 8 8 00 0 0 8 8 8 0 0 00 0 0 0 8 0 0 0 0.b) (5 po ints) Find a basis for the image of t he matrix A.Problem 6) (10 points) Inverse and CoordinatesLetS ="1 1−1 1#andA ="1 1−1 1#.a) (4 points) Find the inverse of S by row reducing the 2 × 4 matrix [S|I2].b) (2 points) Assume the matrix A describes the transformat ion in the standar d basis. Find thematrix B in the basis given by the column vectors of S.c) (2 points) Two of the three matrices A, B, S are similar. Which ones?Problem 7) (10 points) Expectation, Variance, CovarianceThe vectors X =123157and Y =11111−2can be seen as random variables over a probability spacewith 6 elements. They encode two different data measurements, where 6 probes were taken.a) (2 points) Find the expectations E[X], E[Y ].b) (2 po ints) Find the variances Var[X] and Var[Y ].c) (2 points) Find the standard deviations σ[X] and σ[Y ].d) (2 po ints) What is the covariance Cov[X , Y ] of X and Y ?e) (2 points) Find the correlation Corr[X, Y ] of X and Y .Problem 8) (10 points) Combinatorics and Binomial Distributiona) (2 points) We throw 20 coins. What is the probability that 5 heads show up?b) (3 po ints) We throw 7 dice. What is the probability that 3 of the dice show the number 5?c) (5 points) Assume that the random variable X counts the number of heads when throwing acoin 20 times. What is the expectation of this random variable?Problem 9) (10 points) Bayes formulaa) (7 points) We throw 10 coins. What is the probability that the first coin is head if we knowthat 5 times heads comes up?b) (3 po ints) Is the following argument correct? Whether your answer is ”yes” or ”no ”, give ashort reason. The chance that an earth quake hit is P[A] = 1/1000. The chance that a tsunamihits is P[B] = 1/1000. Therefore, the chance that an earth quake and a tsunami hit both isP[A] × P[B] = 1/1′000′000. The event that a quake a nd tsunami hit at the same time is t hereforea one in a million event.Problem 10) (10 points) Data fittingFit the following data using functions of the form f(x) = ax + bx3.x y1 01 11 2-1