Prove that n, nth roots of unity form a group under multiplication.

7. Describe how to achieve the best score by Smith-Waterman in linear space.

Use the formula for computing future value using compound interest to determine the value of an account at the end of 5 years if a principal amount of ​$5,000 is deposited in an account at an annual interest rate of 3​% and the interest is compounded monthly.

2. New uses of polynomials are being discovered frequently. In fact, Tamás Erdélyi, a Texas A & M professor made significant contributions as recently as 1995.

a. Find another mathematician who advanced the study of polynomials and share what you learned. Be sure to read through your classmates' postings first, duplicate facts will not count.

Explain what is meant by deductive reasoning. Give an example of the Law of Detachment and the law of Syllogism.

1 Use Cramer’s rule to determine the solution of the system of equations:

4x1 + 3x2 + 2x3 = 8

−x1 + 2x3 = 12

3x1 + 2x2 + x3 = 3

2. Consider the following system of equations involving a parameter k:

(2 − k)x1 + kx2 = 4

kx1 + (3 − k)x2 = 3

(a) For what value(s) of the parameter k does the system have a unique solution?

(b) Solve the system of equations using Cramer’s rule.

For each of the statements (i)—(iii), state whether it is true or false (i) Every system of linear equations has at least one solution. (ii) A system of four linear equations with three variables always has infinitely many solutions. (iii) One can determine whether two straight lines in R 3 intersect by solving an appropriate system of linear equations. (iv) Given any matrix A, then its reduced row echelon form is not unique. (v) Every elementary row operation can be undone by an(other) elementary row operation. (vi) One of the elementary row operations is to delete a row (i) A system of 3 linear equations with 4 variables cannot have a unique solution. (ii) It is possible to obtain two different reduced row echelon matrices from a given matrix by using two different sequences of elementary row operations. (iii) Elementary row operations on an augmented matrix do not change the solution set of the associated system of linear equations. (i) The matrix ? ? 1 0 a 0 100 0 0 0 7 ? ? is invertible for any value of a. (ii) If a system of linear equations with a square coefficient matrix A has infinitely many solutions, then det(A) = 0. (iii) Elementary row operations do not change the determinant of matrices. (i) A 2 × 2-matrix can have three distinct eigenvalues. (ii) If 0 is an eigenvalue of a square matrix A, then A is not invertible. (iii) Every 3 × 3-matrix can be diagonalized using elementary row operations.

prove that where a,b,c,d,e are real numbers with (a) not equal to zero. if this linear equation ax+by=c has the same solution set as this one : ax+dy=e, then they are the same equation.

Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W ) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T ]AB is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

A young couple buying their first home borrow $65,000 for 30 years at 7.4%, compounded monthly, and make payments of $450.05. After 5 years, they are able to make a one-time payment of $2000 along with their 60th payment.

(a) Find the unpaid balance immediately after they pay the extra $2000 and their 60th payment. (Round your answer to the nearest cent.)
$  

(b) How many regular payments of $450.05 will amortize the unpaid balance from part (a)? (Round your answer to the nearest whole number.)
payments

(c) How much will the couple save over the life of the loan by paying the extra $2000? (Use your answer from part (b). Round your answer to the nearest cent.)
$

Let ? be an eigenvalue of a matrix A. Explain why dim(?) ? 1

Problem 3. How many arrangements of MATHEMATICS are there that have ALL of the following properties:

TH appear together in this order and E appears somewhere before C?