Cartesian Tensors and transformation matrix

[Hacky]

I was just reading chapter on Cartesian tensors and came across equation for transformation matrix as function of basic vectors. I just do not get it and cannot find a derivation. I am too old to learn Latex, I uploaded a word document with the equation. Thanks, Howard

[uiulic]

Howard,

Lij=cos(e'i,ej)=e'i dot ej.

[Hacky]

I am looking at page 930-931 Of Riley, Hobson, Bence.

They go from:

e'j,= Sij ei

X'i = (S^{-1})ij Xj

Define L as inverse of matrix S

X'i = Lij Xj, since rotations of coordinate axes are rigid, transformation matrix L is orthogonal, thus the inverse transformation is

Xi = Lji X'j and

Lik Ljk = \deltaij and Lki Lkj = \deltaij

"furthermore, in terms of basis vectors of the primed and unprimed Cartesian coordinate system, the transformation matrix is given by

Lij = e'i dot ej

I understand the cosine formula for dot product but do not see how the transformation matrix follows from this argument. I am starting to get the Latex, but all the i's and j's are of course subscripts.

Thanks

[uiulic]

using x'ie'i=xjej ; If you start from e'j,= Sij ei, then Sij is some matrix which is defined.x'je'j=xiei or x'jSij ei=xiei ,i.e. Sij x'j=xi, i.e.x'j=[Sinverse]jixi , noting how matrix operation is applicable[again e.g.by dotting by dotting e'k for both sides then you get x'k=xjLkj]...

Some property Lki Lkj = delij, should be remembered once for all.Then you can get everything else.