Advanced Linear Algebra - Lecture 8: Change of Bases

Introduction

• Lecturer: Nathan Johnson
• Topic: Change of Bases in Vector Spaces
• Objective: Understanding how to convert coordinate vectors when changing bases in a vector space.

Basic Concepts

• Coordinate Vectors: Lists of numbers representing vectors in terms of a specific basis.
• Problem: How to represent a vector in a different basis?

Change of Basis Matrix

• Definition: A matrix that converts coordinate vectors from basis B to basis C.
• Notation: Change of basis matrix from B to C denoted by ( P_{B \to C} ).
• Structure: An (N \times N) matrix formed by using vectors from basis B represented in basis C as columns._

Theorem on Change of Basis Matrix

• Property A: Converts coordinate vectors from basis B to basis C.
• Property B: The inverse of the matrix converts vectors back from basis C to basis B.
  • The matrix is invertible.
• Uniqueness: The change of basis matrix is unique for given bases.

Notation Explanation

• Notation (P_{B \leftarrow C}) uses a backward arrow to align subscripts, ensuring adjacent matches in expressions.
• Remembering the Definition: Represent old basis vectors (B) in the new basis (C)._

Proofs

• Proof of Theorem:
  • Part A: Demonstrated using block matrix multiplication.
  • Part B: Proved the invertibility and uniqueness of the matrix.

Example: Change of Basis Matrix Computation

• Objective: Find a change of basis matrix from B to C for polynomial vector space (P_2).
• Steps:
  1. Find coordinate vectors for basis B in terms of C.
  2. Use inversion technique to simplify calculations.
  3. Multiply original coordinate vectors by change of basis matrix to get vectors in basis C.

Alternative Method

• Using Helper Basis: Convert to a helper basis (often the standard basis) to simplify calculations.
• Theorem: If you have three bases (B, C, E as a helper), compute easier change of basis matrices and combine them.
• Procedure:
  1. Compute change of basis matrices from B and C to E (standard basis).
  2. Use reduced row echelon form to find the change of basis matrix from B to C.

Example: Converting Between Ugly Bases

• Objective: Convert between two non-standard bases.
• Steps:
  1. Introduce standard basis E.
  2. Compute change of basis matrices to E.
  3. Use row reduction to find the change of basis matrix between B and C.
  4. Calculate new coordinate vector using the computed matrix.

Summary

• Change of Basis Matrix: Converts vectors between different coordinate representations.
• Inversion Technique: Useful for simplifying calculations.
• Helper Basis Method: Provides a systematic way to handle complex conversions.

• Next lecture will cover linear transformations between arbitrary vector spaces.