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A.1.12 Transformation Laws for Cartesian Components of Vectors and Tensors
and be two orthogonal bases in a Cartesian coordinate system. could be made to coincide with through a rigid body rotation (i.e., a transformation that preserves vector length and angles), where . Since , the components of are direction cosines between and .
Reflection about plane, .
For any vector , its components with respect to and are and , respectively. Using the above relation,
or In matrix form, or Here and are matrices of the same vector, expressed in two different coordinate systems. This is not the same as , where is the linear transformation of by .
Now consider a tensors . Its components with respect to and are given by and , respectively. Thus, , or In matrix form, , or Equivalently, we can show that or . As for vectors, we note that and are the matrices of the same tensor , with respect to two different coordinate systems. This is not the same as .