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Let be a rotation by angle about the origin.
For arbitrary , where is a basis in and is a basis in .
is the coordinate vector of w.r.t so
For some , we can represent as .
, so what is ?
Matrix representation theorem
If and are ordered bases for vector spaces and respectively, then corresponding to each linear transformation there is an matrix such that
is the matrix representing relative to the ordered bases and .
In fact, , where for .
For and is a basis in , find the matrix representing w.r.t. ordered bases (standard 3D basis) to
, where is a basis for . Find .
, where .
forms basis for , and forms basis for
Reversing Linear Transformations
Given and are bases for and respectively,
If is the matrix representing w.r.t. and , then
for , where
If is the matrix representing the linear transformation w.r.t. and , then the rref of is .
Basis is ,
Basis is , ,
What is w.r.t. and ?
- ↑ The correspondence between and given by ; and between and is called isomorphism