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Coordinate Transformation

  1. Oct 11, 2007 #1
    If we want to transform vector A from cooedinate ei to ei',
    then this formula occur:
    Aj' = aij Ai
    But I have a question, if I have found all components of Aj', then I want to transform it back to Ai, what should I do?
    I have tried Ai = aij Aj'
    but it won't give me the same number.
  2. jcsd
  3. Oct 11, 2007 #2


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    Staff Emeritus
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    If, for example, your space is n-dimensional, then, given a particular coordinate system, each point can be written as an array of n numbers (a "vector"). The set of numbers aij, changing from one coordinate system to another can be written as a vector (and, assuming both are "valid" coordinates systems so they have the same dimension as the space, the matrix is non-singular). Then the transformation back the opposite way is just the inverse of that matrix.
  4. Oct 11, 2007 #3
    I am still confused.
    Can you give me an example please?
    Let's say e1' = (2 e1 + 2 e2 + e3) /3
    e2' = 1.4142 (e2 + e2)
    e3' = 0.4714 (e1 + e2 + 4 e3)
    and I have a vectro t = 10 e1 + 10 e2 - 20 e3
    Can you transform it to e1', e2', and e3'?
    I have done that, but when I rewrite it back to the original coordinate, it won't be like that.
  5. Oct 11, 2007 #4
    By the way, I found that t' = 6.667 e1 + 47.14 e3
    Is that OK?
  6. Oct 11, 2007 #5
    You know that [tex]a'_{i} = A_{ij}a_{j}[/tex]. You want [tex]a_{i} = \ldots[/tex]. How can you get rid of the [tex]A_{ij}[/tex] matrix on the right hand side using other matrices?

    If you were working with just numbers and had b=ka, how would you work out a in terms of b? What's the matrix version of this?
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