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1. One set v is a linear combination of u. Prove u is linearly dependent

Yes. My Monday lecture did not spoke about bases or dimensions. But we did take about linear combination and linearly independent sets as well as a vector space.
2. One set v is a linear combination of u. Prove u is linearly dependent

How can I do it without any theorems about bases and dimensions of linear spaces?

4. One set v is a linear combination of u. Prove u is linearly dependent

What is the other route? I might easier to understand.
5. One set v is a linear combination of u. Prove u is linearly dependent

I am an typo. I gave the wrong inequality symbol. Sorry
6. One set v is a linear combination of u. Prove u is linearly dependent

Hi Everybody, I am having some difficulties on the prove this problem. I picked a nice example when I was trying to think about the proof. Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...
7. I Factoring Matrices with Elementary Row Operations

I checked with my calculator. My sequence is the same as the matrix given.
8. I Factoring Matrices with Elementary Row Operations

Dear Everybody, I have some trouble with this problem: Finding a sequence of elementary matrix for this matrix A. Let ##A=\begin{bmatrix} 4 & -1 \\ 3& -1\end{bmatrix}##. I first used the ##\frac{1}{4}R1##-> ##R1##. So the ##E_1=\begin{bmatrix} \frac{1}{4} & 0 \\ 0& 1\end{bmatrix}##. So the...
9. I Finding a Rational Function with data (Pade approximation)

Polynomial interpolation would works well in this data. But pade approximate would better because the error would be smaller. The best situation is having a defined function that does not happen in many cases in application problems. It is usual given by data points.
10. I Finding a Rational Function with data (Pade approximation)

Because I am learning about pade approximation and I am presenting this method to my colleagues, I need to know where to start. I understand that I can do the polynomial interpolation.
11. I Finding a Rational Function with data (Pade approximation)

Dear Everybody, I need some help understanding how to use pade approximations with a given data points (See the attachment for the data). Here is the basic derivation of pade approximation read the Derivation of Pade Approximate. I am confused on how to find a f(x) to the data or is there a...
12. Showing that S is unbounded by contradiction

no. I think that your condition is not possible I can pick a N bigger than R...
13. Showing that S is unbounded by contradiction

I talked to my professor about this question. He recommended me to do proof by contradiction on this question. I need to practice this method. It is good practice for me to do so. I know that the sequence needs move away from the closed ball...
14. Showing that S is unbounded by contradiction

I am confused....are we still using proof by contradiction?
15. Showing that S is unbounded by contradiction

A set S is bounded if there is a closed ball B(r,p)=\{(x,y)\in \mathbb{R}^2: |(x,y)-p|\le R}\ such that B(r,p) is a subset of S. Based by the assumption, I am "given a R" but i can't know the "R".
16. Showing that S is unbounded by contradiction

Dear Everyone, I am attempting a proof of contradiction for this problem. I am stuck on next step. My attempt: Assume that ##S## is bounded. Choose a ##N=\text{greatest integer function of} R+1##... Here is where I am stuck. I want to show that the sequence is moving away from the closed ball...
17. A Prime Factorization Theorem and Number Systems

Another one is the a 2×2 matrix that has all the same integer entries. Will not have a fundamental theorem of arithmetic, I think the only unique factorization is a matrix of -1 or 1. There is a paper in the MAA Journal math monthly for more information about matrix number theory.
18. Proving the preimage of a singleton under f is a singleton for all y

Surjectivity and interjectivity
19. Proving the preimage of a singleton under f is a singleton for all y

The professor was typing quickly and forget then statement. If #f^{-1}({y})# is a singleton for all #y \in Y#, then #f# is bijection.
20. Proving the preimage of a singleton under f is a singleton for all y

Dear Everyone, I have some trouble how to start the proof of this statement. I need to prove the preimage of the singleton under f is the subset of singleton of x and vice versus. My attempt is this: Given y. So we know that definition of the preimage is when all #x# is in #X# , then #f(x)...
21. Geometry Recommendation for High School Geometry book

Dear Everyone, So I would like some recommendation for high school geometry books that are affordable and preferably e-books. Why do I need some books on high school geometry? I would like to improve my geometric reasoning. When I took high school geometry a decade and half ago, I was...
22. Prove rare trig identity

My question was meant for the person who posted the homework question, sorry for the confusion.
23. Prove rare trig identity

do you know sum or difference identities for tangent ?
24. Using a Problem Solving Technique for a simple word problem

I exclude geometry because it is a proof writing course before calculus. I agree with your assessment.
25. Using a Problem Solving Technique for a simple word problem

Thank you for your explanation of why I was wrong with my reasoning.
26. Using a Problem Solving Technique for a simple word problem

I got a negative number as my solution. The back of the book, its positive.
27. Using a Problem Solving Technique for a simple word problem

I think this was an example for me and the other tutors to follow these steps in order to help the intermediate algebra students when I was undergraduate school. I agree with you that this is a common algebra skills. I also struggles with word problems. And I want to remedy that struggle.
28. Using a Problem Solving Technique for a simple word problem

Summary:: I have a document which was written by previous tutor supervisor/instructor at my undergraduate school. I want to learn how to use this technique for other math/physics subject besides algebra... I will provide an example of how to use this technique. I found an old textbook (College...
29. Write the Standard Form of the Equation for this Circle

So what is the formula, then?
30. Write the Standard Form of the Equation for this Circle

Do you know the midpoint formula?