For my math class I am required to write an essay on the following topic:

Write an essay about the existence and uniqueness of a solution to an ODE( ordinary differential equation) ?

I do not require that the essay be written but can you possibly give me some hints and extra information that I may include in my essay.

a = [3, -4, 7] b = [-6, 9, 8] c = [4, 0, 8] d =[7, 1, 7] e = [3, -5, 2, 1] f =[5, -7, -3, 6] g = [3, -4, 4, 3]
P = Projection of
ex. C = |g|(gf/gf) C = gf/|f|
ex. P g --> f = Cgf = C(gf/f) (1/|f|) (f) =( gf/ff)(f)
Find
a. Pg --> f
b. Pa --> 3b + e

Find (cross multiply)
a. ||a X b||
b. ||g X e||
c. ||d X b||
d. Calculate d X b
e. Calculate c X d

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for loop to denote each entry of the vector y.) Your M-file should perform a check on the dimensions of the input variables A and x and return a message if the dimensions do not match. Call the file myrowproduct.m. Note that this le will NOT be the same as the myproduct.m example. Test your function on a random 5x2 matrix A and a random 2x1 vector x. Compare the output with A*x. Repeat with a 3x5 matrix and a 5x1 vector and with a 3x5 matrix and a 1x5 vector. Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it.

5. Recall that if A is an m n matrix and B is a p q matrix, then the product C = AB is denoted if and only if n = p, in which case C is an m q matrix.

(a) Write a function M-file that takes as input two matrices A and B, and as output produces the product by columns of the two matrix. For instance, if A is 3x4 and B is 4x5, the product is given by the matrix C = [A*B(:,1), A*B(:,2), A*B(:,3), A*B(:,4), A*B(:,5)]The function file should work for any dimension of A and B and it should perform a
check to see if the dimensions match (Hint: use a for loop to denote each column of C). Call the file columnproduct.m.Test your function on a random 5x3 matrix A and a random 3x5 matrix B . Compare the output with A*B. Repeat with 3 x 6 and 6 x 4 matrices and with 3 x 6 and 4 x 6 matrices.Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it.

(b) Write a function M- file that takes as input two matrices A and B, and as output produces the product by rows of the two matrices. For instance, if A is 3 x 4 and B is 4x5, the product AB is given by the matrix C = [A(1,:)*B; A(2,:)*B; A(3,:)*B] The function file should work for any dimension of A and B and it should perform a check to see if the dimensions match (Hint: use a for loop to denote each row of C). Call the file rowproduct.m. Test your function on a random 5x3 matrix A and a random 3x5 matrix B . Compare the output with A*B. Repeat with 3 x 6 and 6 x 4 matrices and with 3 x 6 and 4 x 6 matrices. Use the command rand to generate the random matrices for testing.
Include in your lab report the function M-file and the output obtained by running it.

Let p and q be two linearly independent vectors in R^n such that ||p||_2=1, ||q||_2=1 . Let A=pq^T+qp^T.

determine the kernel, nullspace,rank and eigenvalue decomposition of A in terms of p and q.

Solve for the exponential Fourier series f(x) = cos(pix) for 0<x<1.

Sketch the frequency spectrum for the series on the interval -3<= n<= 3.

Suppose f maps a closed and bounded set D to the reals is continuous. Then the Uniform Continuity Theorem says f is uniformly continuous on D. To prove this, we will suppose it is not true, and arrive at a contradiction.

So, suppose f is not uniformly continuous on D. If f is not uniformly continuous on D, then there exists epsilon greater than zero and sequences (a_n) and (b_n) in D for which |a_n -b_n| < 1/n and |f(a_n) - f(b_n)| > epsilon. Give comment as to why this must be true if f is not uniformly continuous on D.

Nothing says the sequences (a_n) and (b_n) converge. But, because f is continuous on D, there exists subsequences of (a_n) and (b_n), say (a_nk) and (b_nk) respectively, that do converge. Give comment as to why this would be true.

Show that both limits of the subsequences must lie in D and that they are in fact equal. Explain why this is leads to a contradiction. Finish the proof.

Let G be a group. Consider the set G with a new operation ∗ given by a ∗ b = ba. Show that (G, ∗) is a group isomorphic to the original group G. Give an explicit isomorphism.

Given the following binary relations R on two sets, for each relation:

• Draw the arrow diagram of R.
• Is R a function, and why?
• If R is a function, determine if it is injective or surjective. Is the function bijective? Justify your answers.

1. 1. R = {(a, 3), (c, 1)} on domain {a, b, c} and codomain {1, 2, 3}
2. 2. R = {(1, a), (3, c), (2, b)} on domain {1, 2, 3} and codomain {a, b, c}
3. 3. R = {(a, b), (b, c), (d, b), (c, c)} on domain {a, b, c, d} and codomain {a, b, c, d}
4. 4. R = {(a, y), (b, z)} on domain {a, b} and codomain {x, y, z}

The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 35 students, requires 3 ​chaperones, and costs ​$ 1,200 to rent. Each van can transport 7 ​students, requires 1​ chaperone, and costs ​$ 90 to rent. Since there are 280 students in the senior class that may be eligible to go on the​ trip, the officers must plan to accommodate at least 280 students. Since only 30 parents have volunteered to serve as​ chaperones, the officers must plan to use at most 30 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation​ costs? What are the minimal transportation​ costs?

How do you solve this word problem?

T is a tree graph on 10 vertices, each is labled with an integer from 1-10. There are four leaves: 1, 2, 3, and 4.

1. What is the number of possible trees, such that the vertices labeled 1, 2, 3, and 4 are leaves, and the other labeled vertices can be either a leaf or not.

1. What is the number of possible trees, such that only 1, 2, 3, and 4 are leaves.

A pizza place is having a special: 10 medium pizzas for $50. They only allow you to choose one of their 6 different specialty pizzas though, with no substitutions. You and 6 friends are having a get-together (so 7 people altogether), and you decide to order pizza for the group. Each person gets to choose one type of pizza, and then you are responsible for choosing the remaining 3. Once all 7 people have chosen their pizza, how many total ways are there for you to complete the order?

Use Euler method (as explained on the white board) to solve numerically the following ODE:

dy/dt=y+t. y(0)=1

You can select the step size. Choose n=3. solve this by hand on a paper and with the aid of Matlab.

Let f(x) = x - R/x and g(x) = Rx - 1/x

a) Derive a Newton iteration formula for finding a root of f(x) that does not involve 1/x_n. To which value does the Newton iterates x_n converge?

b) Derive a Newton iteration formula for finding a root of g(x) that does not involve 1/x_n. To which value does the Newton iterates x_n converge?

Reflect on the concept of polynomial and rational functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest polynomial and rational function you can imagine? In your day to day, is there any occurring fact that can be interpreted as polynomial and rational functions? What strategy are you using to get the graph of polynomial and rational functions?

The Learning Journal entry should be a minimum of 400 words and not more than 750 words.