How Do You Reverse a Vector Transformation?

[R3DH34RT]

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.
Thanks...

 

[HallsofIvy]

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 a_ij, 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.

 

[R3DH34RT]

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.
Thanks...

 

[R3DH34RT]

By the way, I found that t' = 6.667 e1 + 47.14 e3
Is that OK?
Thanks...

 

[AlphaNumeric2]

You know that a'_{i} = A_{ij}a_{j}. You want a_{i} = \ldots. How can you get rid of the A_{ij} 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?