Changing basis in 3D vector space

[WilliamC]

I am trying to work through multiple Schaums outline books on my own including Schaum's Outline of Differential Geometry.

 

I cannot seem to get how to change basis in 3D vectors.

 

From chapter 1, problem 1.47,

 

Given vectors u₁, u₂, and u₃ form a basis in E3 and v₁ = -u₁ + u₂ -u₃, v₂ = u1 +2u₂ - u₃, v₃ = 2u₁ + u₃, show that v₁, v₂, v₃ are linearly independent and find the components of a = 2u₁ - u₃ in terms of v₁, v₂, and v₃.

END OF PROBLEM.

 

 

 

Now that v₁, v₂, v₃ are linearly independent is obvious since there is a zero coefficient for u₂ in the equation for v₃, and I can show that the determinate of the matrix formed by the row vectors (-1, 1, -1), (1, 2, -1), (2, 0, 1) is not equal to zero.

 

I get that the matrix A = [a_ij] i,j, = 1,2,3 are the coefficients a₁₁ = -1, a₁₂ = 2, a ₁₃ = -1, a₂₁ = 1, a₂₂ = 2, a₂₃ = -1, a₃₁ = 2, a₃₂ = 0, a₃₃ = 1 and that there is a matrix B = [b_ij] which will have the coefficients allowing the expressions of the equations u₁ = b₁₁v₁ + b₁₂v₂ +b₁₃v₃, u₂ = b₂₁v₁ + b₂₂v₂ + b₂₃v₃, u₃ = b₃₁v₁ + b₃₂v₂ + b₃₃v₃.

 

What I can't get is how to determine the values of the [b_ij]'s.

 

I've tried cross multiplying matrix A by the 3 x 3 identity matrix to get 3 sets of equations and setting them all equal to zero, which seems to make sense, but i can't get the same answer in the book.

 

Any suggestions? Please, this is driving me insane and I can't get past it to chapter two and I need to also get back to making progress in Schaum's Outline of Tensor Calculus, Outline of Linear Algebra, and Outline of Group Theory too.

 

The answer is supposed to be u₁ = -2v₁ + v₂ -v₃, u₂ = 3v₁ -v₂ + 2v₂, u₃ = 4v₁ -2v₂ +3v₃.

 

 

 

P.S. typing all the formatting takes too much time, wish I could more easily communicate with someone more easily.

Edited March 20, 2012 by WilliamC

[timo]

a seems to be a vector by definition. Hence, it is not clear to me what [math]a_{ij}[/math] and A are supposed to be. It would probably become clear by "reverse-engineering" what you wrote. But I have the feeling that not having defined your mathematical objects may hint to where your problem lies.

 

Hint: [math] \vec a = \sum_i x^i \vec u_i = \sum_{ij} x^i {\bf 1}_i{}^j \vec u_j = \sum_{ijk}x^i \underbrace{A^{-1}{}_i{}^j A_j{}^k}_{={\bf 1}_i{}^k} \vec u_k = \sum_{ijk}\underbrace{x^i A^{-1}{}_i{}^j}_{=y^j} \underbrace{A_j{}^k \vec u_k}_{=\vec v_j} =y^j \vec v_j = \vec a [/math], where A is a suitable transformation matrix, 1 is the identity matrix, and x, y are the coefficients of vector a in the basis of the u and v, respectively. If you're unfamiliar with the index notation, a notation closer to the one you seem to use would be [math]{\bf a} = \vec x \vec {\bf u} = \vec x A^{-1} A \vec {\bf u} = \vec y \vec{\bf v}[/math] (where now bold stands for a vector and the arrow merely indicates a tuple). The important point is only that you get the idea, anyways.

Edited March 20, 2012 by timo

[WilliamC]

OK, see where I confused you and myself.

 

With the vector a = 2u₁ - u₃ the matrix A I described should relate to the vector a, but it doesn't.

 

Question, is it possible to solve the problem without knowing anything about vector a, in which case matrix A I previously described would be the coefficients of vectors u₁, u₂, u₃ and matrix B the coefficients of vectors v₁, v₂, v₃?

 

Or do I have to consider matrix A to be the row vector a and then a new matrix, C which are the coefficients of v₁, v₂, v₃ when expressed as

u₁ = c₁₁v₁ + c₁₂v₂ + c₁₂v₃, u₂ = c₂₁v1 + c₂₂v₂ + c₂₃v₃, u₃ = c₃₁v₁ + c₃₂v₂ + c₃₃v₃?

 

As for the tensor notation I am starting to learn it but I am not proficient with it yet. Perhaps if I can work through this problem in matrix notation then I can redo it in tensor notation, and I know for anything more than 3D I will need to be able to understand the tensor notation, but I'd like to understand it in terms of matrices first.

 

But it just doesn't leap out at me and I can't find anyone who knows anything about it to talk to and as difficult as it is to have to type all this (I don't even know where you get the summation symbols and such, I can write things I can't express with a keyboard) it seriously irritates me but I can't figure it out on my own.

 

So much so I finally broke down and sought out this resource. Thanks for helping.

Edited March 20, 2012 by WilliamC

[Cap'n Refsmmat]

If you'd like to know how timo wrote out his mathematics, just click on it -- a popup will appear giving the syntax for you to write it out yourself. There's also a link to a tutorial.

[WilliamC]

Learning as I go. Thanks.

 

OK, see where I confused you and myself.

 

With the vector a = 2u₁ - u₃ the matrix A I described should relate to the vector a, but it doesn't.

 

Question, is it possible to solve the problem without knowing anything about vector a, in which case matrix A I previously described would be the coefficients of vectors u₁, u₂, u₃ and matrix B the coefficients of vectors v₁, v₂, v₃?

 

Or do I have to consider matrix A to be the row vector a and then a new matrix, C which are the coefficients of v₁, v₂, v₃ when expressed as

u₁ = c₁₁v₁ + c₁₂v₂ + c₁₂v₃, u₂ = c₂₁v1 + c₂₂v₂ + c₂₃v₃, u₃ = c₃₁v₁ + c₃₂v₂ + c₃₃v₃?

 

As for the tensor notation I am starting to learn it but I am not proficient with it yet. Perhaps if I can work through this problem in matrix notation then I can redo it in tensor notation, and I know for anything more than 3D I will need to be able to understand the tensor notation, but I'd like to understand it in terms of matrices first.

 

But it just doesn't leap out at me and I can't find anyone who knows anything about it to talk to and as difficult as it is to have to type all this (I don't even know where you get the summation symbols and such, I can write things I can't express with a keyboard) it seriously irritates me but I can't figure it out on my own.

 

So much so I finally broke down and sought out this resource. Thanks for helping.

 

Let me try again.

 

With matrix B defined as the matrix of coefficients of vectors u₁, u₂, u₃ (what I mistakenly called matrix A originally) , then would C = [B^T]^-1, that is the inverse of the transpose of B? When I work out the result I get something frustratingly close to the correct answer but not quite, and sometimes the Schaum's outline series has incorrect answers so I don't know if my calculations are off or I'm completely on the wrong idea.

 

I don't know how to write matrices, but I'm taking the inverse of B^T by calculating the 9 minors, constructing a 3 x 3 matrix of these values, and multiplying by det B^T.

 

Am I on the right track? I've lost more than a week on this problem and I have to get past it.

[DrRocket]

 

Any suggestions? Please, this is driving me insane and I can't get past it to chapter two and I need to also get back to making progress in Schaum's Outline of Tensor Calculus, Outline of Linear Algebra, and Outline of Group Theory too.

 

 

Trying to do differential geometry, group theory, linear algebra, and tensor calculus simultaneously is neither realistic nor logical.

 

You need to understand linear algebra thoroughly before you undertake differential geometry or tensor calculus. In fact you need a good deal more to study those subjects, including basic real analysis and topology.

 

Also, Schaum's outllines are not the best way to study advanced subjects. They are intended as supplements to other material, a set of lectures or a text, and tend to emphasize symbol pushing over fundamental understanding. This is apparent in the nature of your questions. You might do better to read some real books. A very good book on linear algebra, suitable for study of analysis and geometry is Finite Dimensional Vector Spaces by Paul Halmos.

[WilliamC]

Trying to do differential geometry, group theory, linear algebra, and tensor calculus simultaneously is neither realistic nor logical.

 

You need to understand linear algebra thoroughly before you undertake differential geometry or tensor calculus. In fact you need a good deal more to study those subjects, including basic real analysis and topology.

 

Also, Schaum's outllines are not the best way to study advanced subjects. They are intended as supplements to other material, a set of lectures or a text, and tend to emphasize symbol pushing over fundamental understanding. This is apparent in the nature of your questions. You might do better to read some real books. A very good book on linear algebra, suitable for study of analysis and geometry is Finite Dimensional Vector Spaces by Paul Halmos.

 

I have an embarrassing abundance of resources and books, and I've gotten some ways into linear algebra, but I really don't have much choice as I am not in any position to take classes anymore. I just have to cram in as much as possible while I can.

 

At least the Schaum's outline series provides me with problems, and only by solving problems do I learn.

[Bignose]

At least the Schaum's outline series provides me with problems, and only by solving problems do I learn.

 

Then I suggest you do the problems from the Schaum's, and learn the subject from a more thorough text. Those Schaum's are great as 'quick reference' books -- books you turn to remind yourself of the exact form of the equations so you don't miss a term or change a constant. But, they are not good teaching books. That was never their intent. I think that a real text will get you far farther along.

[WilliamC]

Well I got a few intro linear algebra texts and their solution manuals, and I suppose I have to get past more of this before I can get deeper. But it's so frustrating not being able to solve a problem, as if I am defeated somehow. And as I'm sure you all are aware the number of people in the real world willing to even talk about math is quite small, and of those finding someone able to help me outside a University is impossible.

 

I can't get as good an understanding of everything else I read without being able to deal with more advanced math than calculus and basic differential equations, and I was hoping I could learn tensors and differential geometry and sort of pick up the linear algebra I needed as I went.

 

yea, right.

Edited March 21, 2012 by WilliamC