A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

here

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    find the dual basis for (1,0,0),(0,1,0),(0,0,1)

  2. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Hmm... I am not immediately sure how to solve this. But, let me look into it a moment...

  3. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    OK, I think I've found a method... just confirming.

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  5. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    First, this is the notation I've seen: \[e_i\] is the Cartesian vector, while \[e^i\] represents the dual space vector.

  6. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I'm not entirely sure why the dual space is important, but it seems to be a generalization of Cartesian coordinates into other spaces.

  7. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The method I've found is as follows: 1. We are given the Cartesian vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. 2. We wish to find three corresponding dual vectors. Call them (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3). 3. According to the definition of a dual space, these vectors can be found by examining three sets of equations. 4. Set 1: To find (x1, y1, z1). We take the following equations (1,0,0) * (x1,y1,z1) = 1 (0,1,0) * (x1,y1,z1) = 0 (0,0,1) * (x1,y1,z1) = 0. 5. Set 2: To find (x2, y2, z2). We take the following equations (1,0,0) * (x2,y2,z2) = 0 (0,1,0) * (x2,y2,z2) = 1 (0,0,1) * (x2,y2,z2) = 0 6. Set 3: To find (x3, y3, z3). We take the following equations (1,0,0) * (x3,y3,z3) = 0 (0,1,0) * (x3,y3,z3) = 0 (0,0,1) * (x3,y3,z3) = 1. Notice how we always use the three Cartesian vectors in each set. Also, within each set we are solving for the same Dual vector. And the equations are set to either 0 or 1 depending on the Dual vector we are solving for.

  8. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    In this case, we will get a very trivial dual basis, namely, (x1,y1,z1) = (1,0,0) (x2,y2,z2) = (0,1,0) and (x3,y3,z3) = (0,0,1) which is the same format as we started with.

  9. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I suppose I should add that the * used in my calculations is the typical dot product.

  10. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  11. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I have another example if you want to look at that problem to see if you understand it.

  12. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Question #2: Find the dual basis of {(1,0,0), (1,1,0), (1,1,1)}. We can look at this one if you want. :)

  13. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    well, give me another example and what is the answer here beause it seems we end up having same problem as our result

  14. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    OK. Yes, that is the tricky part of the question you asked me. The answer just happens to be the same as the starting set of vectors. We got lucky in your question. :)

  15. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    In the example I will give you, you will see the difference. :)

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  17. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So, this time the three Cartesian vectors are (1,0,0) , (1,1,0), and (1,1,1).

  18. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    We want to find the dual basis for these three vectors.

  19. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So, we need three dual vectors. We don't know what they look like yet, so call them (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3).

  20. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Do you follow so far?

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  22. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Good! :) So, we will need three sets of equations. For the first dual vector (x1,y1,z1), we need the following set of equations: (1,0,0) * (x1,y1,z1) = 1 (1,1,0) * (x1,y1,z1) = 0 (1,1,1) * (x1,y1,z1) = 0

  23. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wait

  24. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    OK. Let me know if you have a question. :)

  25. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i figured out thst we can still use transpose just like a matrix . and the answer will be (1,1,1),(0,1,1) and (0,0,1) ... am i correct?

  26. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Here's a more detailed look at what I did. 1st Cartesian Vector * 1st Dual Vector = 1 2nd Cartesian Vector * 1st Dual Vector = 0 3rd Cartesian Vector * 1st Dual Vecotr = 0 The equations should always be set to equal 0, except for when the Cartesian Vector and Dual vector are the same position (so, 1st and 1st, 2nd and 2nd, etc.)

  27. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ok, let me read your post...

  28. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    OK, close. You used matrix multiplication, but what we are actually doing is dot product multiplication. Each equation should yield a single number.

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  30. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Did my last "more detailed look" post above make sense before I show you what I did?

  31. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am lost because its not clear, so, the 1 and 0 come from position, explain more

  32. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    So, let's take a quick break from our current set of equations. I agree that the process is not very clear. It's a big jump in logic. Let me show you the big picture as best I can...

  33. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok sir, thank you

  34. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Here are ALL of the equations that we must solve (in word form): 1st Cartesian Vector * 1st Dual Vector = 1 2nd Cartesian Vector * 1st Dual Vector = 0 3rd Cartesian Vector * 1st Dual Vecotr = 0 1st Cartesian Vector * 2nd Dual Vector = 0 2nd Cartesian Vector * 2nd Dual Vector = 1 3rd Cartesian Vector * 2nd Dual Vecotr = 0 1st Cartesian Vector * 3rd Dual Vector = 0 2nd Cartesian Vector * 3rd Dual Vector = 0 3rd Cartesian Vector * 3rd Dual Vecotr = 1 Pay close attention to where the = 1 appears.

  35. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    This big picture will be the same for every problem that asks you to solve for 3 dual vectors.

  36. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  37. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Does that help you see the pattern in the placement of the =1's?

  38. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok, my head is getting hot

  39. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol

  40. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I understand... it's not easy. lol :)

  41. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do you have a pdf?

  42. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I can make one. :)

  43. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok, thanks , i presume not now but can you make it today?

  44. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sure, I will type something up within about, oh... 20 minutes or so. I think I understand the process well enough to make an algorithm for you quickly. :)

  45. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    attach it here

  46. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thanks

  47. jtvatsim
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Alright, will do. Your welcome. :)

  48. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.