Let ∼ be the relation on P(Z) defined by A ∼ B if and only if there is a bijection f : A → B. (a) Prove or disprove: ∼ is reflexive. (b) Prove or disprove: ∼ is irreflexive. (c) Prove or disprove: ∼ is symmetric. (d) Prove or disprove: ∼ is antisymmetric. (e) Prove or disprove: ∼ is transitive. (f) Is ∼ an equivalence relation? A partial order?

1. You plan to take out a $700,000 mortgage for 30 years with monthly payments.
   1. If you don’t have a down payment, they will offer you an interest rate of 5.7%. If you can make a down payment of at least $20,000 they will lower the interest rate to 5.2%. Calculate the monthly payment for each loan.
   2. 1. You would like to see how long it will take you to come up with a down payment. You already have $6,000 and can put away $500 every quarter at 4% interest compounded quarterly. How long will it take you to save up to $20,000?

1. You take out a $480,000 mortgage for 30 years at 5.5% interest compounded monthly. Create an amortization schedule below for the first 3 periods of payment.

1. How much do you pay in total for the loan?

2. How much of this goes toward interest?

(3) Consider the linear map ? : R2 → R2 which sends (1, 0) ↦→ (−3, 5) and (0, 1) ↦→ (4, −1).

(a) What is the matrix of the transformation? What is the change of coordinates matrix? Do they agree? How come?

(b) Where does this transformation send the area between the vector (4, 2) and the x-axis? Explain algebraically and draw a picture.

(c) What is the image of the lower half plane under ?? Explain algebraically and draw a picture.

(d) What is the pre-image of the upper half plane under ?? Explain algebraically and draw a picture.

(e) What is the image of the unit circle under ?? Explain algebraically and draw a picture.

(f) Deduce whether or not linear transformations preserve angles/areas/curves/shapes, etc.

Linear algebra confusion: We haven't learned about rank and such:
For the following I am looking for examples or checking to see why something is impossible.
a.) What does it mean when we have an inconsistent 2x3 linear system
-What if we reversed it to 3x2? Even put other numbers in place.
b.) Can a 2x3 linear system have a unique solution
-What if we reversed it to 3x2? Even put other numbers in place.
c.) Can a 3x2 linear system have infinitely many soluitons
-What if we reversed it to 2x3? Even put other numbers in place.

We had just one day of class and we didn't discuss: inconsistent, unique, infinitely many solutions.
Where can I also reference to learn about this.

2. LetT:R3→R3suchthatT(1,1,—1)=2,1,0]andT(0,1,1)=1,1,1.
(a) Find T( 5,3-7).
(b) Can you find T (1,0,0) from the information given in part (a)? Explain.
(c) Suppose you are now told that T(1,0,0) = 4,2,0. Find the (standard) matrix of T and
compute T(x,y,z). Find a basis for range(T), the range of T.

The metric space M is separable if it contains a countable dense subset. [Note
the confusion of language: “Separable” has nothing to do with “separation.”]
(a) Prove that R^m is separable.
(b) Prove that every compact metric space is separable.

(8) TRUE/FALSE: Circle either T or F. No justification is needed.

(a) (T : F) Each line in R n is a one-dimensional subspace of R n .

(b) (T : F) The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (−1)r , where r is the number of row interchanges made during row reduction from A to U.

(c) (T : F) Adding a multiple of one row to another does not affect the determinant of a matrix.

(d) (T : F) det(A + B) = det(A) + det(B).

(e) (T : F) If the columns of A are linearly dependent, then det A = 0.

(f) (T : F) det AT = (−1) det A.

(g) (T : F) The determinant of A is the product of the diagonal entries in A.

(h) (T : F) If det A is zero, then two rows or two columns are the same, or a row or a column is zero.

(i) (T : F) If two row interchanges are made in succession, then the determinant of the new matrix is equal to the determinant of the original matrix.

Let H be the subset of all skew-symmetric matrices in M_3x3

a.) prove that H is a subspace of M_3x3 by checking all three conditions in the definition of subspace.

b.) Find a basis for H. Prove that your basis is actually a basis for H by showing it is both linearly independent and spans H.

c.) what is the dim(H)

Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1 ∈ S13 belong to the alternating group A13? Justify your answer

Use the Cauchy Criterion to prove the Bolzano–Weierstrass Theorem, and find the point in the argument where the Archimedean Property is implictly required. This establishes the final link in the equivalence of the five characterizations of completeness discussed at the end of Section 2.6.

Recall that a standard 52-card deck has four suits, ♥, ♦, ♣, and ♠, each of which has13cards, one each of the following kinds, A, K, Q, J,10,9,8,7,6,5,4,3,and2. A hand of seven (7) cards is drawn at random from such a deck. (This means that you get the cards as a group, in no particular order, and with no possible way of getting the same card twice in the hand.) Find the probability that the hand . . .

1. 2. 3.

4. 5.

... is a flush, i.e. all the cards in the hand are from the same suit. [1]

. . . has four cards of the same kind. [1]

. . . has exactly three cards of one kind, two cards of another kind, and two cards of yet another kind. [1]

. . . has cards of seven different kinds. [1]

... is a straight, i.e. a set of cards that can be arranged to be consecutive with no gaps in the sequence AKQJ1098765432, where we allow the sequence to wrap around the end. (So 3 2 A K Q J 10 would count as a straight, for example.) [1]

Show all your work!

Find the infinite series solution about x = 0 for the following DE, using Bessel's, Legrende's, or Frobenius method equations.

3x^2y" + 2xy' + x^2y = 0

Find the first 4 non-zero terms in the series expansion. Do not use k=n substitutions

How do you imagine the graph of a three variable system of three equations would look? How about the graph of a system of four equations in four variables?
First, think about a two variable system, the corresponding graph and what each equation in the system represents. Then, expand this idea to three variables and discuss the possible cases for solutions to the system. Finally, extend the idea to four variables.
If you produce three items that each require three inputs, then you can use a system of three equations in three variables to solve. As an example, you could make small , medium and large pizzas ( these would be your variables). Your equations ( constraints) would come from limitations on how much dough , pizza sauce and cheese you have.
How can you relate a point, a line segment, a square and a cube?
How can you relate a point, a line segment , a circle and a sphere? Hint, start with the smallest of these and think about how you could build up to the next one and then the next one.

It is college allgebra.