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- anonymous

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

- jtvatsim

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

- jtvatsim

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

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## More answers

- anonymous

ok

- jtvatsim

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

- jtvatsim

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

- jtvatsim

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.

- jtvatsim

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.

- jtvatsim

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

- anonymous

ok

- jtvatsim

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

- jtvatsim

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. :)

- anonymous

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

- jtvatsim

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. :)

- jtvatsim

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

- anonymous

ok

- jtvatsim

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

- jtvatsim

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

- jtvatsim

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).

- jtvatsim

Do you follow so far?

- anonymous

yes

- jtvatsim

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

- anonymous

wait

- jtvatsim

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

- anonymous

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?

- jtvatsim

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.)

- jtvatsim

Ok, let me read your post...

- jtvatsim

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

- anonymous

ok

- jtvatsim

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

- anonymous

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

- jtvatsim

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...

- anonymous

ok sir, thank you

- jtvatsim

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.

- jtvatsim

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

- anonymous

ok

- jtvatsim

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

- anonymous

ok, my head is getting hot

- anonymous

lol

- jtvatsim

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

- anonymous

do you have a pdf?

- jtvatsim

I can make one. :)

- anonymous

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

- jtvatsim

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. :)

- anonymous

attach it here

- anonymous

thanks

- jtvatsim

Alright, will do. Your welcome. :)

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