Expressing Linear Operators as Matrices: A Comprehensive Guide

In Linear Algebra, the concept of linear operators is fundamental. A linear operator maps a vector space V onto itself, and every such operator can be represented by a matrix, under certain conditions. This article provides a detailed exploration of how to express linear operators as matrices and explains the underlying principles in an accessible way.

Understanding Linear Operators

An operator (L) is defined as a linear operator if it satisfies the following properties:

For any vectors (u, v in V), we have (L(u v) L(u) L(v)). For any scalar (lambda) and vector (u in V), we have (L(lambda u) lambda L(u)).

In simpler terms, a linear operator preserves the linear structure of the vector space.

Finite-Dimensional Vector Spaces and Bases

A finite-dimensional vector space (V) over a field (F) (such as the real or complex numbers) can be spanned by a finite number of vectors. Any set of (n) vectors in (V) is a basis if they are linearly independent and span (V). This means that any vector in (V) can be expressed as a linear combination of these basis vectors.

Matrix Representation of Linear Operators

Given a linear operator (L) and a basis (A [a_1, a_2, ..., a_n]) of the vector space (V), we can express the action of (L) on the basis vectors. Specifically, the operator (L) applied to each basis vector (a_i) can be written as (L(a_i) lambda_{i1}a_1 lambda_{i2}a_2 ... lambda_{in}a_n).

Notation and Representation

Using the notation (Lambda_i [lambda_{i1}, lambda_{i2}, ..., lambda_{in}] in F^n), we can express the action of (L) on the basis (A) as:

(L A^T [L(a_1), L(a_2), ..., L(a_n)]^T)

This can be interpreted as (L A^T [Lambda_1, Lambda_2, ..., Lambda_n]^T A^T), where (L_A) is the matrix representation of (L) in the basis (A).

Matrix Equality

By arranging the vectors and their images properly, we obtain the matrix equality:

(L A^T L_A A^T)

This equality represents the action of the operator (L) on the vectors in the basis (A) and can be computed using matrix multiplication.

Change of Basis

When the basis changes, the matrix representation of the operator also changes. If a new basis (B) is introduced, the matrix (L_B) corresponding to the operator (L) in the new basis can be found using a transformation matrix (T):

(L_B^T T L_A^T)

Here, (T) is a non-singular matrix that transforms vectors from the old basis to the new basis. This transformation ensures that the operator remains well-defined regardless of the choice of basis.

Conclusion

The ability to express linear operators as matrices is a powerful tool in Linear Algebra. This representation simplifies many computational tasks and provides a clear understanding of the operator's action on the vector space. Understanding the relationship between operators and their matrix representations is crucial for advanced studies in mathematics and its applications in various fields.