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anonymous
 one year ago
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anonymous
 one year ago
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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0find the dual basis for (1,0,0),(0,1,0),(0,0,1)

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Hmm... I am not immediately sure how to solve this. But, let me look into it a moment...

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1OK, I think I've found a method... just confirming.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1First, this is the notation I've seen: \[e_i\] is the Cartesian vector, while \[e^i\] represents the dual space vector.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1I'm not entirely sure why the dual space is important, but it seems to be a generalization of Cartesian coordinates into other spaces.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1The 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.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1In 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.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1I suppose I should add that the * used in my calculations is the typical dot product.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1I have another example if you want to look at that problem to see if you understand it.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Question #2: Find the dual basis of {(1,0,0), (1,1,0), (1,1,1)}. We can look at this one if you want. :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, give me another example and what is the answer here beause it seems we end up having same problem as our result

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1OK. 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. :)

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1In the example I will give you, you will see the difference. :)

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1So, this time the three Cartesian vectors are (1,0,0) , (1,1,0), and (1,1,1).

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1We want to find the dual basis for these three vectors.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1So, 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).

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Do you follow so far?

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Good! :) 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

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1OK. Let me know if you have a question. :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i 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?

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Here'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.)

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Ok, let me read your post...

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1OK, close. You used matrix multiplication, but what we are actually doing is dot product multiplication. Each equation should yield a single number.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Did my last "more detailed look" post above make sense before I show you what I did?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i am lost because its not clear, so, the 1 and 0 come from position, explain more

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1So, 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...

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Here 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.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1This big picture will be the same for every problem that asks you to solve for 3 dual vectors.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Does that help you see the pattern in the placement of the =1's?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok, my head is getting hot

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1I understand... it's not easy. lol :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok, thanks , i presume not now but can you make it today?

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Sure, 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. :)

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Alright, will do. Your welcome. :)
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