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

a.) suppose Ax=0. Show that A^TAx=0.

b.)Suppose A^TAx=0. show Ax=0.

c.) by part a and b, we can conclude that Nul(A) = Nul(A^TA), and thus dim(Nul A) = dim(Nul(A^TA)), and thus nullity(A) = nullity(A^TA). prove the columns of A are linearly independent iff A^TA is invertible.

The following matrix is the augmented matrix for a system of linear equations. A =

(a) Write down the linear system of equations whose augmented matrix is A.

(b) Find the reduced echelon form of A.

(c) In the reduced echelon form of A, mark the pivot positions.

(d) Does the system have no solutions, exactly one solution or infinitely many solutions? Justify your answer

Show that the sequence an = (−1)^n doesn’t converge to 1 nor −1. Can it converge to anything other than 1 and −1?

we are given n chips which may be working or defective. A working chip behaves as follows: if we connect it to another chip, the original chip will correctly output whether the new connected chip is working or is defective. However, if we connect a defective chip to another chip, it may output any arbitrary answer (defective---->might say the other one is working /defective).

In the class, we saw that if strictly more than half the chips are working, then there is an algorithm that finds a working chip using O(n) tests.

1) Prove that even when we only have a single working chip and a single defective chip (i.e.,n = 2), there is no algorithm that can find the working chip in general.

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . . , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set. (Remark: How do you tell if a set is infinite??)

Find the Laplace transform of d²y/dt²

i) A set of 4 6-tuples (“sextuplets”) is linearly independent: (always), (never), (sometimes). ii) A set of 6 4-tuples (“quadruplets”) is linearly independent: (always), (never), (sometimes). iii) A set of 4 equations with 6 unknown variables which is consistent has a unique solution: (always), (never), (sometimes). iv) A set of 4 equations with 6 unknown variables is inconsistent: (always), (never), (sometimes) v) A set of homogeneous equations is inconsistent: (always), (never), (sometimes) vi) The solution to a set of homogeneous equations is unique: (always), (never), (sometimes)

For a 2 by 2 invertible matrix A, define the condition number to be cond(A) = ||A|| ⋅ ||A||^-1. Assume that the matrix norm is defined using the Euclidean vector norm.

(a) Find two 2by2 invertible matrices B and C such that cond(B + C) < cond(B) + cond(C).

(b) Find two 2by2 invertible matrices B and C such that cond(B + C) > cond(B) + cond(C).

(c) Suppose that A is a symmetric invertible 2by2 matrix. Find cond(2A) and cond(A2) in terms of cond(A).

(d)do the results from part (c) hold if A is not symmetric? You can either prove the results, or find counterexamples.

Use Laplace Tranform in solving the ff.:

After cooking for 45 minutes, when a cake is removed from an oven, its temperature is measured at 300°F. 3 minutes later, its temperature is 200°F. The oven is preheated, and so at t=0, the cake mixture is at the room temperature of 70°F. The temperature of the oven increases linearly until t=4 minutes, when the desired temperature of 300°F is attained; thereafter the oven temperature is constant 300°F for t is greater than or equal to 4 minutes.

Solve the following:

a.) devised a mathematical model for the temperature of a cake while it is inside the oven and after it is taken out of the oven.
b.) how long will it take the cake to cool of to a room temperature of 70°F?

Prove that \strongly connected" is an equivalence relation on the vertex set of a directed graph

Is 100202345X a valid ISBN number? If not, what would the correct check digit have to be ?

Solve the congruence 121x ≡ 5 mod 350.

The more compounding periods per year, the lower the effective rate of return. T or F

The stated interest rate is the real or true rate of return on an investment T or F

Compound interest yields considerably higher interest than simple interest. T or F

Interest is the rental fee charged by a lender to a business or individual for the use of money. T or F

Exact interest method uses 365 days as the time factor denominator in the simple interest formula T or F

The total payback of principal and interest is known as compound amount of a loan. T or F

Prove the following for undirected graphs:
(a) A 3-regular graph must have an even number of vertices.
(b) The average degree of a tree is strictly less than 2.