Advanced Linear Algebra, Lecture 2.7: Change of basis

Advanced Linear Algebra, Lecture 2.7: Change of basis

If T is a linear map from X to U, and X≅R^n and U≅R^m, then we can define bijections B from X to R^n, and C from R^m, sending our bases of X and U to the standard bases. This defines a linear map N=CTB^{-1} from R^n to R^m, and hence a basis. If X=U, then this means that C=B, and N and T are similar. Along these lines, similar matrices can represent the same linear map but with respect to a different choice in basis. If B=P^{-1}AP, then P is a "change of basis matrix", and we see an explicit example of how to construct this in the 2x2 case, and the generalization to larger matrices should be apparent. Thought this lecture, we rely on commutative diagrams to illustrate these similarity transforms.

[Jaylooker]

The commutative diagram with the change of basis at 7:24 reminds me of homology

[MrSmackdonkey]

thanks diagrams helpfull, would be nice to include subscript notations for “with respect to” like in book jim hefferon linear algebra page 267 or notation for coordinate representation as used in FIS or roman