Lecture - 5.1 Change of Basis

Understanding Change of Basis in Vector Spaces

Introduction to Vector Spaces and Linear Transformations

• The discussion begins with the association of column representations for vectors in a vector space V based on an ordered basis, and how linear transformations between two vector spaces V and W can be represented by matrices when bases are fixed.

Exploring Changes in Basis

• The focus shifts to understanding how the column representation of vectors changes when the ordered basis is altered, as well as how the matrix representing a linear transformation varies with different bases.

Identity Transformation and Basis Representation

• A finite-dimensional vector space V is considered with an ordered basis beta , alongside a new basis beta' . The goal is to explore the relationship between column representations of a vector under these two bases.

Relationship Between Column Representations

• To analyze this relationship, the identity linear transformation I_V: V to V is introduced. This transformation serves as a natural point of reference for examining how representations change across different bases.

Change of Basis Matrix Definition

• The matrix corresponding to the identity transformation with respect to both bases beta and beta' , denoted as I_V,beta,beta', becomes crucial for understanding how these representations relate mathematically.

Invertibility of Change of Basis Matrix

• It is established that the change of basis matrix I_V,beta,beta' is invertible because it corresponds to an invertible linear transformation (the identity).

Dimensions and Properties of Change of Basis Matrices

• The change of basis matrix from one basis to another has dimensions equal to the dimension of vector space V. An exercise suggests that switching from base β' back to base β, results in an inverse relationship between their respective change matrices.

Example: Calculating Change of Basis Matrix

Practical Calculation Using Standard Bases

• A practical example using standard basis vectors in R^2: let’s denote them as e1 = (1, 0), e2 = (0, 1), while considering another set defined by vectors (1, 1), (1, -1).

Evaluating Identity Transformation at Different Bases

• By evaluating what happens under the identity transformation at each standard basis vector, we derive columns for our new representation which leads us towards calculating our change-of-basis matrix explicitly.

Verifying Relationships Between Representations

Brute Force Verification Methodology

• Given any arbitrary vector expressed in terms of standard coordinates, we can compute its representation under both bases directly through substitution into their definitions.

Confirming Consistency Through Calculations

• After performing calculations on specific examples like v = (2, 3), we find consistent relationships between its representations across both bases confirming our earlier theoretical findings about change-of-basis matrices.

Linear Operators and Their Matrix Representations

Transitioning from Vector Representation to Linear Operators

• Moving forward, attention turns toward understanding how matrices representing linear operators behave when changing from one ordered basis to another within a single vector space context.

Similarity Transformation Concept Introduction

• A key concept emerges: two matrices are said to be similar if there exists an invertible matrix such that one can be transformed into another through multiplication by this invertible matrix.

Importance in Further Studies

• This exploration lays foundational knowledge critical for deeper studies into linear algebra properties shared among similar matrices. Future discussions will include concrete examples illustrating these principles further.