Bases Transformation

Consider two bases (e1,e2) and (˜e1,˜e2), where we consider the former the old basis and the latter the new basis.

Each vector (˜e1,˜e2) can be expressed as a linear combination of (e1,e2):

˜e1=e1S11+e2S21  ˜e2=e1S12+e2S22

(1.0) is the basis transformation formula, and the object S is the direct transformation {Sji,1≤i,j≤2}, (assuming a 2x2 matrix) which can also be written in matrix form:

[˜e1˜e2]=[e1e2][S11S12 S21S22] =[e1e2]S

The resulting matrix is the direct transformation matrix from the old basis to the new basis. Note that the rows of S appear as superscripts and the columns appear as subscripts.

Converting in the reverse direction (new to old) requires doing (1.1), but with the inverse matrix

[e1e2]=[˜e1˜e2][S11S12 S21S22]−1 =[˜e1˜e2][T11T12 T21T22] =[˜e1˜e2]T

where T is the inverse transformation matrix, or T=S−1.