Linear Maps — Studies in Algebra and Group Theory

5.0 Linear Maps

Having established the structure of vector spaces, we now study the morphisms between them. Just as group homomorphisms preserve the group operation, linear maps preserve both vector space operations.

Definition 5.0.1 (Linear map). Let and be vector spaces over . A linear map (or -linear map, or homomorphism of vector spaces) is a function satisfying:

for all (additivity);
for all and (homogeneity).

Equivalently, preserves all linear combinations:

Note. A linear map is completely determined by its values on a basis. If is a basis for , and we specify for each , then for any : This is the universal property of bases: specifying a linear map from is equivalent to choosing where to send the basis vectors.

Definition 5.0.2 (Space of linear maps). The set of all linear maps from to is denoted When , we write and call elements endomorphisms.

Proposition 5.0.1 ( is a vector space). The set is itself a vector space over , with operations:

Addition: for all ;
Scalar multiplication: for all .

Proof. We must verify that and are linear maps when are.

For : Let and .

For : Similarly,

The vector space axioms for follow from those of (operations are defined pointwise). The zero vector is the zero map , and additive inverses are . ∎

Note. The verification that and are linear is “rather boring, but worth checking if you’re not sure.” The operations on are inherited pointwise from , so the axioms follow routinely.

Theorem 5.0.1 (Dimension of ). If and , then

The proof of this theorem will emerge naturally when we establish the correspondence between linear maps and matrices in the next section.

Note. We will soon see: if and , then . In terms of bases for and , linear maps become matrices!

5.0.1 Dependence on the scalar field

The notions of basis, dimension, and linearity depend crucially on the choice of scalar field . The same set can be a vector space over different fields, with different dimensions.

Example 5.0.1. as a vector space over different fields: Consider as a vector space:

Over : , with basis . Every complex number is .
Over : , with basis . Every complex number is the -linear combination .

Note. More generally, if is a subfield (e.g., or ), then any -vector space is automatically a -vector space by “restriction of scalars”—we simply forget that we can multiply by elements of . The -dimension is always at least as large as the -dimension.

Proposition 5.0.2 (Linearity over subfields). If and are -vector spaces and is a subfield, then:

is a -vector space (by restriction of scalars).
Any -linear map is also -linear.
The converse fails: not every -linear map is -linear.

Thus in general.

Example 5.0.2. Complex conjugation is -linear but not -linear: Consider the map

-linearity: For and :

✓
(since for ) ✓

Not -linear: For and : Since , complex conjugation is not -linear.

This example illustrates a profound point: the “natural” maps between spaces can depend on what scalars we consider. Complex conjugation is an -linear automorphism of that is not -linear—it does not commute with multiplication by .