Let T and S be linear transformations of a vector space V, and TS=ST

(a) Show that T preserves the generalized eigenspace and eigenspace of S.

(b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)² (t-3)²?. What is the jordan canonical form of S.

(c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.

show that a 2x2 complex matrix A is nilpotent if and only if Tr(A)=0 and Tr(A^2)=0. give an example of a complex 2x2 matrix which is not nilpotent but whose trace is 0

If equilateral triangles are constructed on the sides of any triangle, prove that the distances between the vertices of the original triangle and the opposite vertices of the equilateral triangles are equal.

5a. Write the equation of the polynomial with leading coefficient 3 that has degree 4, with x-intercepts at -2 and 1, both having a multiplicity of 2.

b. Write the equation of the polynomial which goes through the following points. (-2,0),(-1,0),(5,0), and (0,10).

c. (1+i) is a zero of f(x)=x^4 - 2x^3 - x^2+ 6x - 6. Find the other zeros

Please show all the work. Expert and correct answers only.

A video game designer decides to include a character in a new game that only moves along a parabolic path. The designer uses the function x²+50+10 to model this character’s path, where y is the height of the character in pixels and x is the time in seconds that it takes the character to move across one parabolic path.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.




d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this situation? Write 1-2 sentences to explain your answer.


2. A satellite dish has the shape of a parabola, the U-shaped graph of a quadratic function. Suppose an engineer has determined that the shape of one of the satellite dishes offered by the company can be modeled by the quadratic function y=2/27x²-4/3x , where y is the vertical depth of the satellite dish in inches and x is the horizontal width in inches.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.

d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this situation? Write 1-2 sentences to explain your answer.




3. Suppose an investor builds a stock portfolio with a variety of shares in various high tech companies. The value of the stock portfolio is modeled by the function 2x²-20x+100 , where y is the value of the portfolio in hundreds of dollars and x is the time in months.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.




d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this
situation? Write 1-2 sentences to explain your answer.

Show that if F is a finite extension of Q, then the torsion subgroup of F^* is finite

How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 104 km/h?

42.6 m

If, in actuality, a 870–kg car with an initial speed of 104 km/h is observed to coast up a hill to a height 31.9 m above its starting point, how much thermal energy was generated by friction?

9.11×104 J

What is the average force of friction if the hill has a slope 7.7° above the horizontal?????

Hi please give an example and explain using the Matrix Transformations below. Will rate! Thank you.

Topic: Matrix Transformation

Given the matrix transformations T1 and T2, under what real-world conditions will T1 o T2 = T2 o T1?

(7) How might geometry be used to build a feeling of community?

Theorem. If A is an n by n square matrix, then the following statements are equivalent.

1. A is invertible.
2. The system Av=b has at least one solution for every column-vector b.
3. The system Av=b has exactly one solution for every column-vector b
4. The system Av=b has only the trivial solution (0,0,0,...0) ( Homogeneous systems).
5. The reduced row-echelon form of A is the identity matrix.
6. A is a product of elementary matrices.

Can you please help me to give some short examples or short proofs or logical arguments for EACh statements. EACH statements from 1-6 please.

Sylvia is considering depositing $600 at the end of each semi-annual period, for 5 years earning interest of 8%. She would like to know how large a one-time lump sum deposit she could make, at the same rate, to have the same amount of money after 5 years.

Sylvia later decides needs $ 11,000 in 8 years. She has the opportunity to make a one-time investment that will earn 10% compounded quarterly. How much must she invest to reach her goal?

Develop a real world scenario that illustrates the Order of Operations . Your post should include a mathematical expression that represents your unique scenario, and a step by step explanation of how the Order of Operations is used to find your answer.

INFORMATION FROM THE VIDEO: THE LIZARD DID 8 PUSH-UPS IN 13 SECONDS.

Assumptions:

There are 121 dietary Calories (kilocalories) in 100 grams of crickets

The crickets (that the lizard we see in the video eats) weigh 0.5 grams each

It takes 0.1 dietary Calories of energy for the lizard to do one push-ups

The lizard in the video spends a total of 30 minutes of each day doing push-ups

During his 30 minutes of push-ups, he does them at the same rate that you see in the video

The lizard does these displays every day for 30 days during the breeding season

The question you need to answer is: How many crickets did the lizard need to eat to fuel his 30 days of push-up displays?

To find out how many total push-ups the lizard did during the breeding season, count how many push-ups he did in the video and how long it took him to do them. You now have a push-up count per unit time that you can use to figure out how many push-ups the lizard would do per day and how many he would do in 30 days. (For example, if he did one push-up per minute, then he would be able to do 30 push-ups in 30 minutes. At 30 minutes per day, he would do 900 push-ups in 30 days.)

Calculate how many dietary Calories he spent doing all those push-ups. (Continuing our example, 900 push-ups x 0.1 calories = 90 dietary Calories) Note - you can leave your answers in dietary Calories, no need for conversion.

Calculate how many dietary Calories there are in a cricket.

Calculate how many crickets he would have to have eaten to fuel all those push-ups.

Let X be a finite set. Describe the equivalence relation having the greatest number of distinct equivalence classes, and the one with the smallest number of equivalence classes.