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Linear Algebra : Modules, Linear Operators, Characteristic and Minimal Polynomials, Generators, Abelian Groups and Annihilators
Prove that for all odd integers n, (1^n)+(2^n)+(3^n)...+(n^n) is a proper multiple of 1+2+3+...n
If a,b,c are odd integers, show that all real roots of ax^2+bx+c=0 are irrational numbers.
If x were replaced with its opposite in each of the terms of the given polynomial, will it result in the opposite of the polynomial?
How many rectangles (of all sizes) are there in this grid? Can you generalize your results to an n by n grid? Can you generalize further?
The half-life of a certain radioactive element is 100days. That means that after 100 days ½ of the radioactive substance will have decayed.
Graph the following function when a=3 and b=2. Develop a generic expression (i.e., as a function of "a" and "b") to find the "x" and "y" intercepts.
Planetary orbit [2]. The expression z = ax2 + bxy + cy2 + dx + ey + f is known as a quadratic form. The set of points (x, y) where z = 0 is a conic section.
Walt made an extra $9000 last year from a part-time job. He invested part of the money at 9% and the rest at 8%.
Formulate two word problems from day-to-day life that can be translated to quadratic equations.
Maximum height. If a baseball is projected upward from ground level with an initial velocity of 64 fcct per second, then its height is a function of time.
Maya has deposited $600 in an account that pays 5.64% interest, compounded continuously. How long will it take for her money to double?
Banach space and closed subspace.Let I = [a,b] be a finite interval. Show that the space C(I,Rn) of continuous function
Explain the relevance and application of exponential functions in real-life situations. Explain the relevance and application of logarithmic functions.
Solve the following equation and check your answer. (1/2b)- (1/2)= 1/4b. Solve the following equation 19 - 3n = - 2n.
Solve the following problem by identifying a variable, writing an equation that describes the situation, and then solving the equation.
The total amount to be invested is $40000, and because "The Don" is concerned about risk, he only wishes to invest $13500 in stock funds
Solve a diet plan which is designed for one week (from a chosen restaurant). the person must meet the dietary requirements listed
Pairwise Sequential Voting.A seventeen-member committee must elect one of four candidates
For the line y= 2x+2 find a line parallel to it that passes through the points (2,0) (4,4). Find the slope and intercept for the line 2y-x-4=0.
Symplectic Matrix, Eigenvalues and Multiplicity.Prove that if ? is an eigenvalue of M , then so is ?-1 , and that these have the same multiplicity.
Show that Q/Z is the torsion subgroup of R/Z. This problem is quite straightforward if you use the definitions and stay focussed.
A team of surveyors mark off distances represented by vectors of (85m, 55 degrees), (43m, 0 degrees) and (37m, 345 degrees). Solve for the resultant.
Explain why we cannot square a sum by simply squaring each term of the sum. For example, (a + b)2 is not equal to a2 + b2.
Two identical water tanks labeled A and B are resting at the same elevation and are linked via a pipe with a valve.