1. For a map f : V ?? W between vector spaces V and W to be a linear map it must preserve the structure of V . What must one verify to verify whether or not a map is linear?

2. For a map f : V ?? W between vector spaces to be an isomorphism it must be a linear map and also have two further properties. What are those two properties? As well as giving the names of the properties, explain what the names mean.

3.Every linear transformation is an isomorphism, but the isomorphism f : x y ?? x y is not a linear transformation. Why

Below you are given the description of two quadrilaterals. On a piece of paper (labelled and in order), use the appropriate tools (straightedge, compass, etc.) to construct the described quadrilaterals and the perpendicular bisectors of each of their sides. After each construction, state why there is no circle that will circumscribe the quadrilateral. 1. a parallelogram that is not a rectangle 2. a non-isoceles trapezoid. There is no diagram.

let triangle ABC be a triangle in which all three interior angles are acute and let A'B'C' be the orthic triangle.

a.) Prove that the altitudes of triangle ABC are the angle bisectors of triangle A'B'C'.

b.) Prove the orthocenter of triangle ABC is the incenter of traingle A'B'C'.

c.) Prove that A is the A' -excenter of triangle A'B'C'.

Given A*B*C and A*C*D, prove the corollary to Axion B-4.

Prove the following problems using the complex plane model of Euclidean geometry, in the spirit of Erlangen Program: 1. Prove that the diagonals of a parallelogram bisect each other. 2. Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all the sides of the parallelogram. 3. Prove the Cosine Law for triangles: In a triangle with the sides a, b, and c, the square of the side opposite C = is expressed as c2 = a2 + b2 - 2 b a cos . 4. Prove the theorem: The bisector of an angle of a triangle divides the opposite side into two segments which are proportional to the sides that include the angle.

Prove the following using the triangle inequality:

Given a convex quadrilateral, prove that the point determined by the intersection of the diagonals is the minimum distance point for the quadrilateral - that is, the point from which the sum of the distances of the vertices is minimal.

Plot the following parabolic functions in Excel and then answer the discussion question below . Use at least 8 points for each plot: Each graph is worth 15 points and must include 8 points. 1) y=x2-6x+8 2) y=x2+4 3) y=(-1/2)(x2) 4) What is the maximum number and the minimum number of x-intercepts for a parabola? Why? (15 points) In order to obtain full credit, you will need to plot the functions in Excel or using the Open Office feature and attach the results to your response.

Four people are using a voting weight system to make decisions. They have the weights of 7,5,4,and 2. They use a quota of 13. Compute the Banzhaf power index for the voter of weight 5.

1) Determine the two angles

sin(2θ)=0.9179.

2)Determine the solution sets

12cos^2θ=3

I have no idea of what is happening when the professor explains how they got the angles or soln sets, they automatically know what quadrant it is in and what angle it should be, and what the next angle(s), sets are. I cant imagine it. I don't know how they see that on the unit circle.

Suppose you have a piece of cardboard with length 32 inches and width 20 inches and you want to use it to create a box. You would need to cut a square out of each corner of the cardboard so that you can fold the edges up. But what size square should you cut? Cutting a small square will make a shorter box. Cutting a large square will make a taller box.

Since we haven’t determined the size of the square to cut from each corner, let the side length of the square be represented by the variable x. Write a simplified polynomial expression in x and note the degree of the polynomial for each of the following geometric concepts:

The length of the base of the box once the corners are cut out, the width of the base of the box once the corners are cut out, the height of the box, the perimeter of the base of the box, the area of the base of the box, the volume of the box.

a. Suppose the earth were wrapped tightly in a 25,000 mile belt. Now suppose someone adds 1 mile to the belt. If the belt is raised uniformly above the earth’s surface, how high above the surface will it be? Give your answer in feet. (Guess first, before you calculate this.)

b. This time, suppose someone adds 1 foot to the belt. Again, raise the belt uniformly above the earth’s surface—how high will the belt be? Give you answer in inches.

c. Finally, suppose a regulation NBA basketball is wrapped tightly in a 29.5 inch belt. Now suppose someone adds 1 foot to the belt. If the belt is raised uniformly above the ball’s surface, how high will it be? Give your answer in inches. Are you surprised by this result?

Why do  perpendicular bisectors of the three sides of a triangle all meet at a single point? and also why do  angle bisectors of the three sides of a triangle all meet at a single point ?

1) Find the intervals of increasing and decreasing for f(x) = 2x3 – 4x2.
2) Find the local minimum and maximum points, if any, of                         f(x) = 2x3 – 15x2 + 36x – 14. 3) Find the inflection points, if any, of f(x) = 2x3 – 15x2 + 36x – 14. Give the intervals of concavity upward and downward for f(x). 4) Find the absolute maximum and minimum of f(x)= 2x3 – 15x2 + 36x – 14 on the interval [0,5]. 5) Sketch the graph of y = 2x3 – 15x2 + 36x – 14 using the information from #2-4 along with the intercepts. 6) Given C = .02x3 + 55x2 + 1250, find the number of units x that produces the minimum average cost per unit, ?. ̅ 7) Find the maximum, minimum, and inflection points of f(x) = x4 – 18x2 + 5.

Give a method for solving Fermat's Problem when a triangle has an agle greater than 120°.

Show that in any triangle the angle bisectors are concurrent. The point where they meet is called the incenter of the triangle, and is the center of the incircle, whose radius is the distance from the incenter to any of the sides of the triangle.