Problem 4: Suppose M is a random matrix, and x is a deterministic (fixed) column vector....

Problem 4: Suppose M is a random matrix, and x is a deterministic (fixed) column vector. Show that E[x' M x] = x' E[M] x, where x' denotes the transpose of x.

Expert Solution

milcah answered 1 year ago

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector....

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector. a.) suppose Ax=0. Show that ATAx=0. b.)Suppose ATAx=0. show Ax=0. c.) by part a and b, we can conclude that Nul(A) = Nul(ATA), and thus dim(Nul A) = dim(Nul(ATA)), and thus nullity(A) = nullity(ATA). prove the columns of A are linearly independent iff ATA is invertible.

4. The product y = Ax of an m n matrix A times a vector x...

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for...

1. For an m x n matrix A, the Column Space of A is a subspace...

1. For an m x n matrix A, the Column Space of A is a subspace of what vector space? 2. For an m x n matrix A, the Null Space of A is a subspace of what vector space?

4. The product y = Ax of an m × n matrix A times a vector...

4. The product y = Ax of an m × n matrix A times a vector x = (x1, x2, . . . , xn) T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as defined above (Hint: use...

Vector A has a magnitude of 4 m/s and it is on the positive x axis...

Vector A has a magnitude of 4 m/s and it is on the positive x axis . Vector B has a magnitude of 6 m/s and forms a 30 degrees angle with the positive x axis. Vector C has a magnitude of 8 m/s and forms a 60 degree angle with the negative x axis. Find: a) The magnitude of A + B + C = D Give your answer with two significant figures. b) The angle vector D form...

Suppose C is a m × n matrix and A is a n × m matrix....

Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b. 1. Show that if this system is consistent then the solution is unique. 2. If C = [0 ?5 1 3 0 ?1] and A = [2 ?3 1 ?2 6 10] ,, find x (if it exists) when (a) b =[1...

Write a function that will accept one integer matrix E and one column vector F by...

Write a function that will accept one integer matrix E and one column vector F by reference parameters, and one integer I as a value parameter. The function will return an integer v, which is the i-th coefficient of row vector denoted by E*F (multiplication of E and F). For example, if V = E*F, the function will return v, which is equal to V[i]. Explain the time complexity of this function inside of the function code as a comment....

True/False: If A is an invertible m x m matrix and B is an m x...

True/False: If A is an invertible m x m matrix and B is an m x n matrix, then AB and B have the same null space. (Hint: AB and B have the same null space means that ABX = 0 if and only if BX = 0, where X is in Rn.)

4. Consider the random variable Z from problem 1, and the random variable X from problem...

4. Consider the random variable Z from problem 1, and the random variable X from problem 2. Also let f(X,Z)represent the joint probability distribution of X and Z. f is defined as follows: f(1,-2) = 1/6 f(2,-2) = 2/15 f(3,-2) = 0 f(4,-2) = 0 f(5,-2) = 0 f(6,-2) = 0 f(1,3) = 0 f(2,3) = 1/30 f(3,3) = 1/6 f(4,3) = 0 f(5,3) = 0 f(6,3) = 0 f(1,5) = 0 f(2,5) = 0 f(3,5) = 0 f(4,5) = 1/6...

Suppose that the coefficient matrix of a homogeneous system of equations has a column of zeros....

Suppose that the coefficient matrix of a homogeneous system of equations has a column of zeros. Prove that the system has infinitely many solutions. What are the possibilities for the number of solutions to a linear system of equations? Can you definitively rule out any of these?

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