In Sakurai's Modern quantum mechanics it is said that the rotation matrix in three dimensions that changes one set of unit basis vectors $(x, y, z)$ into another set $(x' , y' , z' )$ can be written as $$\begin{bmatrix} xx' & xy' & xz' \\ yx' & yy' & yz' \\ zx' & zy' & zz' \end{bmatrix} $$ But shouldn't it be the transpose of matrix given above as the transformation matrix is given by coordinates of transformations of bases ?
2026-03-30 12:19:34.1774873174
Confusion regarding transformation matrix
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The given matrix is correct.
What we want is the matrix of the identity map but in a different basis. (We are not changing the vectors, we are just changing the basis).
The way we get this matrix is we write the images of basis vectors ${x,y,z}$ and look at their coordinates in the basis ${x' ,y', x'} $ The coordinates of $x$ become the first column the coordinates of $y$ the second and so on.
The way to determine these coordinates is to take the inner product.
And thus we get the matrix given.