Lecture 12: Advanced Linear Algebra - Changing Basis of a Standard Matrix
Overview
Focus on changing the basis of a linear transformation represented as a standard matrix.
Previously covered representing linear transformations as standard matrices.
Today's goal: learn the process of changing the basis for these matrices.
Key Concepts
Changing Basis for Coordinate Vectors
If you have vector ( v ), create coordinate vector ( v ) with respect to basis ( B ).
To change to another basis ( C ), multiply by a change of basis matrix.
Changing Basis for Linear Transformations
Similar to coordinate vectors, need to multiply on both sides (input and output) by change of basis matrices for standard matrices.
Main Theorem
Describes how to change basis for standard matrices of linear transformations.
Uses pre-and-post multiplication by change of basis matrices.
Setup
Finite dimensional vector spaces ( V ) and ( W ).
Linear transformation ( T ) between these spaces.
Two input bases: old (( B )) and new (( C )).
Two output bases: old (( D )) and new (( E )).
Theorem Statement
If the standard matrix with respect to old bases ( B ) and ( D ) is known, use a formula to construct the standard matrix with respect to new bases ( C ) and ( E ).
Pre-and-post multiply by appropriate change of basis matrices.
Diagrammatic Proof
Shows transitions between different basis representations.
Coordinate vectors in ( V ) and ( W ) using bases ( B, C ) and ( D, E ).
Change of basis matrices transform these vectors between different bases.
Demonstrates the equality of matrices using coordinate transformations.
Algebraic Proof
Involves multiplying a product of matrices by an arbitrary coordinate vector and proving equality of two matrix products.
Example: Transpose Map with a New Basis
Task
Compute the standard matrix of the transpose map with respect to a new basis (the Pauli basis).
Importance
Demonstrates using a theorem to change the basis of a matrix.
Though calculations are involved, results in simpler (diagonal) matrix representations.
Computation
Determine change of basis matrices ( P_{E \leftarrow B} ) and ( P_{B \leftarrow E} ).
Multiply those with the original standard matrix to get the new standard matrix.
Notes
Direct computation from the definition yields the same result.
Simplifies the representation of operations like the transpose map.
Conclusion
Highlights benefits of changing bases, even to non-standard ones, for simpler matrix representations.
Prepares for upcoming topics: properties of linear transformations, range, null space, eigenvalues.