Translate the phrases to a system of linear equations: y= a1*x+b1 and y=a2*x+b2 with a variables, e.g. x and y. What do the y-intercepts represent in your example? What does the solution (or intersection) represent in your example? Solve the system of equation for x and y.

b. Find three rational numbers between 1/9
and 0. 12 overbar

Let A be some m*n matrix. Consider the set S = {z : Az = 0}. First show that this is a vector space. Now show that n = p+q where p = rank(A) and q = dim(S). Here is how to do it. Let the vectors x₁, . . . , x_p be such that Ax₁, . . . ,Ax_p form a basis of the column space of A (thus each x can be chosen to be some unit vector with a 1 corresponding to the position of a column vector that is part of a (maximally) linearly independent set) and let the vectors z1, . . . , zq be a basis for S. Then show that the two sets together i.e. the set {x1, . . . , xp, z1, . . . , zq} form a basis of the n-dimensional Euclidean space.

Using the result above, offer a direct proof of the result r(X′X) = r(X) without appealing to the product rank theorem

axis symmetry, graph max or min value, domain, range and all intervals where function increases or decreases. f(x)=5(x-3)^2+6 f(x)=-7(x-8)^2+1 f(x)=-6(x+4)^-8 f(x)=2(x+2)^2+9

A rectangular piece of metal is 30in longer than it is wide. Squares with sides 6in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 2706incubed3​, what were the original dimensions of the piece of​ metal?

what is the original width?

what is the original length?

In Week 4 we learned about quadratic equations. In physics a quadratic equation can be used to model projectile motion. Projectile motion can describe the movement of a baseball after it has been hit by a bat, or the movement of a cannonball after it has been shot from a cannon. A penny falling from the Empire State Building can even be modeled with this equation!

The projectile motion equation is s(t)=-16t^2+vt+h where s(t) represents the distance or height of an object at time t, v represents the initial speed of the object in ft/s, and h is the initial height of the object, measured in feet.

If an object is starting at rest, then v=0 (such as for a penny being dropped from a building). If the object is starting from the ground, h=0. The baseball or cannonball situations, each have an initial velocity. For example, the initial velocity of the baseball is based on the speed at which the ball comes at you (the speed of the pitch).

Come up with a situation that you can model with this equation. Describe the situation, and tell us what v and h are. Fill in the values so that you have a quadratic equation. If you do research to find initial velocities, include the links to the websites where you found that information. If you would like to make up your own numbers as well, you can (be creative)!

Once you have your equation, find the maximum height as well as the time it takes to reach that maximum. Then use your equation to find when the object hits the ground (i.e. the x-intercepts).

Finally, use those three points as well as the initial height to sketch a graph. You can take a photo of it and include the image, or use an online graphing calculator and take a screenshot if that is easier.

Find the equation of the straight lines and draw the graph.
a) Go through point (2,1) and have slope 5.
b) Go through points (3, -1) and (4,5)

Let A and b be the matrices A = 1 2 4 17

3 6 −12 3

2 3 −3 2

0 2 −2 6

and b = (17, 3, 3, 4) . (a) Explain why A does not have an LU factorization. (b) Use partial pivoting and find the permutation matrix P as well as the LU factors such that PA = LU. (c) Use the information in P, L, and U to solve Ax = b

Determine which of the following sets of vectors form a basis for R 3 . S = {(1, 0, −1),(2, 5, 1),(0, −4, 3)}, T = {(−1, 3, 2),(3, −1, −3),(1, 5, 1)}.

Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional space. Express the operator T(p) = p'

as a matrix (i) in basis {1, x, x 2 }, (ii) in basis {1, x, 1+x 2 } .

Suppose an ice cream cone has negligible thickness and the tip of the cone is identified with origin (0,0,0)and the circular opening at the top is identified with the equation x^2+y^2=1, z=1. if you fill the cone with ice cream "rounded" so that the surface of the sections forms part of the surface x^2+y^2+z^2=2 and the ice cream has variable density f= x^2+y^2+z^2, what is the mass of the ice cream?

A certain business keeps a database of information about its customers. A. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning.

Sketch a graph of the polynomials below as demonstrated in the Polynomial Functions video.  

Note: Do not just graph on a calculator and copy the graph! No credit will be given without supporing work or explanation. You should be prepared to do problems like this without a calculator on exams!

h(x)=x³(x-2)(x+3)²

• List of real zeros and their multiplicities:

• End behavior:

resolve the matrix gauss Elimination

Use the simplex method to solve the linear programming problem.

The maxium P= ____ at (x,y) ____