Find the two scalars (in C) λ1 and λ2 so that A − λI is singular. For

A = [₁^{1 ₁ -1}]

Use the fact that A − λI is singular iff det(A − λI) = 0.

For each λi find a basis for RS(A−λ_iI). Each basis will consist of a single vector, verify that the two vectors you found are orthogonal.

Match each table with its equation.

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

OPTIONS -Linear, Square Root, Quadratic, Absolute Value, Cubic, Reciprocal

1. Plot each point. Label them A,B,C , and D.

A. (2,-2) B. (-4,0 C. (1.5 5/2) D. ),5)

2. Find the x and Y intercepts of the gien equation and write answers in ordered pair form (x,y). Then graph the equation using these intercepts.

4x-3y=12

X-interceptis:______

Y-Intercepts is:_______

3. Find the slope of the line passing through the given points using the slope formula, then graph the line using the slope found. (3,5) and (-5,6)

m=_________

5. Find the slope and the y-intercept of the line with the given equation. Remember you must first solve this equation for y.

2x=3y=6

M=_________

6. Solve the system of equations by the elimination mehtod. If the lines intersect, identify the point of intersection, if the lines are parallel or coincindent, indicate the appropriate solution.

2x=11y=-10

5x=4y=22

7. Use the point-slope form of a linear equation y-y₁=m(x-x_{1 ) to write the equation of the line with the given slope and point. Write your final answer in the slope-intercept form of a linear.equation y=mx+b.}

_{Slopee=2 and passes through (3,5)}

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.