2.3 Linear Maps
In this section, we will discuss linear maps between vector spaces. A linear map is a function that preserves the operations of vector addition and scalar multiplication.
Table of Contents
Introduction
Just like Hassani, we will begin by reviewing common maps in general.
Some examples of maps are:
- The map
defined by . - The map
defined by . - The map
defined by , where and are real-valued functions. - The map
defined by , which represents a rotation by an angle . - The map
defined by , which represents a curve in 3D space parameterized by . We often use this to represent the trajectory of a particle in space over time. - The map
defined by , which represents a translation by the vector .
Let
The set of all linear maps
If the domain and codomain are the same, i.e.
In other words, we can add linear maps and multiply them by scalars, and the result is still a linear map. Let's define these operations.
First, the zero map
Next, the sum of two linear maps
Lastly, the scalar multiplication of a linear map
A linear map
In other words, an isometric map preserves the lengths of vectors and the angles between them.
If
Some examples of linear maps are:
-
For any one-dimensional vector space, the only linear maps are the zero map and scalar multiplication by a constant;
for some . If is an isometry, then must be a complex number with unit magnitude, i.e. for some . -
Let
be a polynomial in (the space of complex polynomials in ). The map defined by is a linear map, since differentiation is a linear operation. However, it is not an isometry, since it does not preserve the inner product. Similarly, the map defined by is also a linear map, but not an isometry. We can write both maps in terms of the basis
(the monomials). can be expressed as for some coefficients . Then, we have -
Rotations and reflections in
and are isometries, since they preserve lengths and angles. For example, the rotation map defined by is an isometry. -
In Minkowski space, the set of all isometries is called the Poincaré group. This group includes translations, rotations, and boosts (changes in velocity). The Poincaré group is important in special relativity, as it describes the symmetries of spacetime.
Two linear maps
Let
Proof.
(
(
Similarly,
Any endomorphism
Proof.
(
(
where we used the assumption that
Let
By Theorem 2.3.7, we conclude that
From what we have learned so far, we can determine if two linear maps
if and only if . if and only if for all vectors in a basis of (Box 2.3.6). if and only if for all and (Theorem 2.3.7).
2.3.1 Kernels
For a linear map
forms a subspace of
Intuitively, if we visualize vectors as arrows on a plane, the kernel is the set of all vectors that get squished down to the zero vector by the linear map
Proof. We need to show that the kernel is closed under addition and scalar multiplication, and that it contains the zero vector.
First, it obviously contains the zero vector, since
Next, let
By the linearity of
Thus,
Finally, let
Thus,
The image (or range) of a linear map
forms a subspace of
Proof. We need to show that the image is closed under addition and scalar multiplication, and that it contains the zero vector.
First, it obviously contains the zero vector, since
Next, let
Since
Finally, let
Thus,
A linear map
If the kernel contains only the zero vector, it is known as a trivial kernel. Injectivity means that different vectors in the domain map to different vectors in the codomain.
Proof.
(
Since
(
Let
Since
All linear, isometric maps
Proof. Suppose that
By the isometric property of
Since the inner product of a vector with itself is zero if and only if the vector is the zero vector, we have
Lastly,
For a linear map
assuming that both vector spaces are over the same field
Proof. Let
The vectors
The number of vectors in this basis is
This completes the proof.
In a finite-dimensional vector space, an endomorphism is bijective if it is either injective or surjective.
Proof. Let
If
Since the rank of
If
This means that
We will skip Example 2.3.15; see Hassani for details.
2.3.2 Linear Isomorphisms
We have alluded to the idea of isomorphisms in previous sections.
In the context of vector spaces, two vector spaces can look (be notationally) different but still be fundamentally identical in structure.
For instance, the vector space of polynomials of degree at most
Let
If
A linear isometry
Proof. By Theorem 2.3.12, we know that
A surjective linear map
Proof.
(
(
If
Proof. Suppose that
From the fact that
Now, consider the linear combination of the images under
Since
If two finite-dimensional vector spaces
Proof. Suppose that
Since
This means that for any
Let
Proof. We can prove this in one line,
Let
If
We need to show that
Then
where we used the linearity of
Thus,
To show surjectivity, let
By Theorem 2.3.20, since
By the dimension formula for quotient spaces, we have
Combining these two equations, we get
which is the dimension theorem.
Generally, we have
Suppose
Lastly, consider the linear map
for all
Thus,
Summary and Next Steps
In this section, we explored various properties of linear maps between vector spaces, including kernels, images, injectivity, and isomorphisms. We established key theorems that connect these concepts, such as the dimension theorem and the characterization of injective maps.
Here are the key points to remember:
- Definition 2.3.2: A linear map is a function between vector spaces that preserves vector addition and scalar multiplication.
- Theorem 2.3.9: The kernel of a linear map is a subspace of the domain.
- Theorem 2.3.10: The image of a linear map is a subspace of the codomain, and its dimension is called the rank.
- Theorem 2.3.11: A linear map is injective if and only if its kernel is trivial (contains only the zero vector).
- Theorem 2.3.13: The dimension theorem relates the rank and nullity of a linear map to the dimension of the domain.
- Theorem 2.3.20: Isomorphic vector spaces have the same dimension.
With these concepts in mind, we will now take a closer look at complex vector spaces and inner product spaces in the next section, which are fundamental in quantum mechanics.