Vector transformation followed by coordinate point substitution and viceversa, both given the same result. Choose the best answer

Vector transformation followed by coordinate point substitution, and vice versa, giving the same result, indicates the consistency and interchangeability of applying linear transformations in algebra. This process can be visualized within the context of linear algebra, particularly when dealing with transformations in spaces like ( mathbb{R}^n ).

Given a vector (mathbf{v}) in ( mathbb{R}^n ) and a linear transformation ( T: mathbb{R}^n rightarrow mathbb{R}^m ), applying ( T ) to (mathbf{v}) and then substituting coordinates (or vice versa) leads to the same result due to the properties of linear transformations. This involves the transformation matrix ( A ) associated with ( T ), which acts on (mathbf{v}) to produce a new vector in ( mathbb{R}^m ).

The process is as follows:

1. Vector Transformation: Apply the transformation ( T ) to vector (mathbf{v}), resulting in ( T(mathbf{v}) = Amathbf{v} ), where ( A ) is the transformation matrix.

2. Coordinate Point Substitution: After applying the transformation, we can substitute the coordinates of (mathbf{v}) into the resulting vector to find its new location in ( mathbb{R}^m ).

The