(1) Complete the matrix A (fill in the two blank entries) so that A has
eigenvectors x1 = (3,1) and x2 = (2,1) :
(2) Find a different matrix B with those same eigenvectors x1 and x2 , and with
eigenvalues λ1 = 1 and λ2 = 0 . What is B10 ?
(1) Given two arbitrary finite numbers f0 and f1 , if fk = fk−1 + fk−2
(2) Suppose B∈ Rm×n . Show that BT B is positive definite if and only if B has
independent columns.
Given a matrix
Find
(1) The adjoint of A .
(2) Find the characteristic polynomial of A .
(3) Find the eigenvalues of A .
(4) Based on the result in 3., is A invertible? If yes, find the inverse of A .
(5) Compute A6 + 2A5 + A4 .
Let Pn be the space of polynomials of degree at most n . Let the linear map
L : P2 → P1 be defined as L( p(x)) = ( p(x) − p(0)) / x . Find the matrix representation
of L with respect to the bases {1+ x, x + x2 , 1− x2} for P2 and {x, 1− x} for P1.
Let N be a geometric random variable with its sample space SN = {1, 2,...} . Find
the conditional pmf P[N is odd | N ≤ m] .
The waiting time T of a customer at a store counter is zero if there is no one else
waiting at the counter, and an exponential random variable with mean 2 if there is
someone waiting at the counter. The probability that someone is waiting when the
customer arrives at the counter is 3/4.
(1) Find the cumulative distribution function (cdf) of T .
(2) Find the mean and variance of T .
Let X be a Gaussian random variable with mean 30 and variance 36. Construct a
Let random variables X and Y have the joint pdf:
(1) Find the marginal pdf of X .
(2) Find the joint cdf of X and Y for x ≥ 1, 0 ≤ y ≤ 1.
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