In two dimensions, consider a reference frame with basis vectors:
.
The reference frame is rotated counter-clockwise through and angle , obtaining a new reference frame with basis .The components Vx and Vy in the basis of a generic vector are transformed according to the rotation matrix
.
1) Express the component of the vector in the basis
2) Express how the sets of basis vectors transform. That is, express the vectors as a function of the vectors and vice-versa.

Question

In two dimensions, consider a reference frame with basis vectors:
 .
 The reference frame is rotated counter-clockwise through and angle   , obtaining a new reference frame with basis  .The components Vx and Vy in the   basis of a generic vector  are transformed according to the rotation matrix 
   . 
1) Express the component of the vector  in the  basis 
2) Express how the sets of basis vectors transform. That is, express the vectors  as a function of the vectors  and vice-versa.

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