A bag contains three red marbles, three green ones, one lavender one, two yellows, and one orange marble. HINT [See Example 7.]

How many sets of five marbles include at least two red ones?

Please show all work and label everything correctly.

1: Consider the sum of the first n positive integers that leave a remainder of 4 when divided by 6. FInd and prove a formula for this sum.

Prove that step by step and clear handwritten. Follow the comment

Conception: Limit point

1.Prove N'=empty

2.Prove Q'=R

3. E=(2,5), E'=[2,5] my question is that why the collection limit point of E includes 2 and 5? obviously, 2 and 5 are not in E. If they can be limit point in E does that mean R can be all set's collection of all limit point? such as R'=R, Q'=R, N'=R???

The conception of limit point

I know the proof , so don't show me the proof and i need you to answer my doubt. follow the comment

1. Q'=R

if I pick a point in Q and I can find infinitely rational number in the neighborhood so p is a limit point, but why it is R. My doubt is that R includes irrational numbers as well but Q doesn;t have irrational numbers, so why R=Q' and R contains all the limit points in Q.

2. For example, if Q'=R, that means R contains all the limit points of Q. So if i pick any point in Q there will have a sequence converges to that point rather than the point itself. For example, if i pick an irrational number in Q, I don't think we can find an irrational number point in Q

Please find how many numbers more than 1,000 and less than 1,000,000 are not multiple of 2, 3, or 5.

use the elimination method to find the general solution for the given linear system where differentiation is with respect to t. 2x'+y'-x-2y=e^-t and x'+y'+2x+2y==e^t

solve differential equation.
y" + 9y = 18t^2 + 9t + 1 ; y(0) = 5/3 ; y'(0) = 10

5. Prove the Following:

a. Let {v1, . . . , vn} be a finite collection of vectors in a vector space V and suppose that it is not a linearly independent set.

i. Show that one can find a vector w ∈ {v1, . . . , vn} such that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that Span(S) = Span(v1, . . . , vn).

ii. Suppose T ⊂ {v1, . . . , vn} is known to be a linearly independent subset. Argue that the vector w from the previous part can be chosen from the set {v1, . . . , vn} \ T.

b. Let V be a vector space and v ∈ V a vector in it. Argue that the set {v} is a linearly independent set if and only if v 6= ~0. Then use this fact together with part i of part a to prove that if {v1, . . . , vn} is any finite subset of V containing at least one non-zero vector, you can obtain a basis of Span(v1, . . . , vn) by simply discarding some of the vectors vi from the set {v1, . . . , vn}.

c. Suppose {v1, . . . , vn} is a linearly independent set in V and that {w1, . . . , wm} is a spanning set in V.

i. Prove that n ≤ m. Hint: use part ii of part a to argue that, for any r ≤ min(m, n), there is a subset T ⊂ {w1, . . . , wm} of size r such that {v1, . . . , vr , w1, . . . , wm} \ T is a spanning set. Then consider the two possibilities when r = min(m, n).

ii. Conclude that if a vector space has a finite spanning set, then any two bases are finite of equal length. (Necessarily, this means that our notion of dimension from class is well-defined and any vector space with a finite spanning set hence has finite dimension).

Hello, I am a bit confused as how line integrals can exist in 2D.

For 2D, we are dealing with two variables: x and y.

For 3D, we are dealing with three variables: x,y and z=f(x,y), such that f(x,y) = x+y. Since line integrals use the x and y components as inputs and f(x,y) = z as the output, should not line integrals exist in 3D always?

solve: y''+8y'+7y=f(t), y(0)=y'(0)=0, expressing your answer in terms of a convolution, using partial fractions for:

i)f(t)=e^t

ii)f(t)=t^1/2

iii) f(t)=2sin(2t)

An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Each subcontractor will have enough time to work on up to two projects during the day. Each project should be completed by exactly two subcontractors. Given below are the distances between the subcontractors and the projects a. Draw a network to represent the possible subcontractor-project assignments. b. Develop a linear model which would be used to minimize total mileage costs. do not understand part b the writing on the other post are not legible for me to understand

Rules for positive definite matrix.

What are they? My text book explains them in a very confusing way.

Also:

"A matrix A is said to be positive definite if it is symmetric and if and only if each of its leading principal sub matrices has a positive determinate"

How do I get the leading principal sub matrices?

A couple has decided to purchase a $150000 house using a down payment of $17000. They can amortize the balance at 11% over 30 years.
a) What is their monthly payment?
Answer = $  1265.78
b) What is the total interest paid?
Answer = $  322680.8
c) What is the equity after 5 years?
Answer = $  
d) What is the equity after 25 years?
Answer = $

Equity is assets minus liability. So you want to find the present value of the loan after 5 years and 25 years, then subtract those from the value of the house, which ignores the down payment