Due to mismatched information, accurate solutions to the student's problems about ordered bases, coordinate matrix, and linear transformation in the subject of mathematics at the college level cannot be provided.
The student's question poses three problems regarding ordered bases, change of coordinates matrices, coordinate vectors, and the effects of linear transformations. The subject is complex mathematics, dealing with abstract algebra and linear transformations.
The provided information seems to suggest questions related to invariance of distances under rotations, change of coordinates, and vector products; however, it's not directly applicable to the student's question. Therefore, an answer to the student's question cannot be accurately provided based on the given information.
To solve the student’s queries, the proper method for finding the change of coordinates matrix from one basis to another would consist of writing each vector of the new basis as a linear combination of the old basis vectors and assembling these into a matrix. The coordinate vector of a polynomial relative to a given basis is found by expressing the polynomial as a linear combination of the basis vectors.
The matrix of a linear transformation with respect to a basis is found by applying the transformation to each basis vector and expressing the results as coordinate vectors relative to the same or another specified basis.