Why is the same linear map represented by different matrices?

A matrix depends on a choice of bases for the domain and codomain. Changing bases conjugates the matrix, so a single linear operator corresponds to a whole similarity class of matrices, which is why canonical forms are useful.

What does the rank-nullity theorem tell you?

It says the dimensions of the kernel and image add up to the dimension of the domain. This immediately decides when a linear system has solutions and how large its solution set is, and when a map is injective or surjective.

Linear Transformation

A linear transformation is a map between vector spaces that preserves addition and scalar multiplication, the morphism of linear algebra represented by a matrix once bases are chosen.

Definition

A linear transformation between vector spaces over the same field is a function that respects vector addition and scalar multiplication, so that the image of a linear combination is the corresponding linear combination of images.

Scope

This topic covers linear maps and their kernels and images, the rank-nullity theorem, the matrix of a linear map relative to bases, change of basis, composition and invertibility, and the correspondence between abstract linear maps and matrices.

Core questions

What does it mean for a map to be linear?
How do the kernel and image measure injectivity and surjectivity?
How is a linear transformation represented by a matrix, and how does that matrix change with the basis?
When is a linear transformation invertible?

Key theories

Rank-nullity theorem: For a linear map between finite-dimensional spaces, the dimension of the domain equals the dimension of the image plus the dimension of the kernel, tying together injectivity, surjectivity, and the solvability of linear systems.
Matrix representation and change of basis: Choosing bases represents a linear map by a matrix, composition corresponds to matrix multiplication, and changing bases conjugates the matrix, so similar matrices represent the same operator in different coordinates.
Isomorphism with matrices: The space of linear maps between finite-dimensional spaces is isomorphic to a space of matrices, making the abstract and concrete viewpoints interchangeable and reducing linear algebra to matrix computation.

Clinical relevance

Linear transformations model rotations, projections, and scalings in geometry and graphics, observables and time evolution in quantum mechanics, and the layers of linear maps inside neural networks. The rank-nullity theorem governs the solvability of every linear system encountered in applications.

History

The matrix calculus of Cayley and Sylvester gave linear maps a concrete representation in the mid-nineteenth century, while Grassmann and Peano provided the abstract, coordinate-free view of linear maps between vector spaces that underlies the modern theory.

Key figures

Arthur Cayley
James Joseph Sylvester
Hermann Grassmann
Giuseppe Peano

Seminal works

hoffman1971
roman2008
lang2002

Frequently asked questions

Why is the same linear map represented by different matrices?: A matrix depends on a choice of bases for the domain and codomain. Changing bases conjugates the matrix, so a single linear operator corresponds to a whole similarity class of matrices, which is why canonical forms are useful.
What does the rank-nullity theorem tell you?: It says the dimensions of the kernel and image add up to the dimension of the domain. This immediately decides when a linear system has solutions and how large its solution set is, and when a map is injective or surjective.