Span, Linear Independence, and Basis

In this section, we will explore the concepts of span, linear independence, and basis in the context of vectors and vector spaces. This has a clear parallel to this page, where we discussed the same topics. However, this page will delve deeper into the concepts and provide a more advanced and abstract understanding of these topics.

Table of Contents

• Table of Contents
• Linear Combinations and Span
  • Proof of Equivalence of Definitions
• Linear Independence
  • Lemma: Span and Linear Independence

Linear Combinations and Span

A linear combination of vectors is a way to combine these vectors by multiplying each vector by a scalar and adding them together. For example, given vectors and , the linear combination is equal to . Formally, we can define a linear combination as:

A linear combination of vectors , where is a vector space over a field , is a vector of the form:

where .

The set of all possible linear combinations of a set of vectors is called the span of those vectors. Intuitively, it describes how far you can reach by moving in different directions defined by the vectors.

There's another way to think about the span of a set of vectors. Let be a subset of vectors. Then, consider all subspaces of that contain - they would contain all possible linear combinations of the vectors in , since they are closed under addition and scalar multiplication. But they can also contain other vectors that are not in . Hence, we can take the intersection of all such subspaces to get the span of :

The span of a set of vectors is the set of all possible linear combinations of these vectors. It is denoted by .

Alternatively, the span of a set of vectors can be defined as the intersection of all subspaces of that contain the set of vectors.

Where is a subspace of .

The two different definitions differ in philosophical approach; the definition from linear combinations is a "bottom-up" approach from the vectors themselves, while the subspace definition is a "top-down" approach from sets bigger than the vectors.

Proof of Equivalence of Definitions

We can prove that the two definitions of span are equivalent. First, let's lay out our definitions:

Let be a set of vectors in a vector space over a field . Then, the span of is defined as:

1. The set of all possible linear combinations of the vectors in :

2. The intersection of all subspaces of that contain :

Generally, in order to show that two sets and are equal, we need to show that and . As such, we need to show that defined by linear combinations is a subset of defined by subspaces, and vice versa.

1. Linear Combinations Subspaces:

   Let be a vector in the span of defined by linear combinations. Then, can be written as a linear combination of the vectors in :

   Since , the vectors in are in the span of . By definition, all the subspaces on the right-hand side of the second definition contain . And since they are subspaces, they are closed under addition and scalar multiplication, meaning any linear combination of vectors in is in the subspace. Hence, for all subspaces that contain .

2. Subspaces Linear Combinations:

   This is a bit more abstract because we don't actually know what the subspaces are, so we cannot construct a concrete value for . Nevertheless, a key insight can help us here.

   Consider some sets , , and , and their intersection . If , then is in all three sets. This means that is a subset of , , and . Then, contains all the elements in , and similarly for and .

   Let's apply that here. The right-hand side with the subspaces contains the intersection of all subspaces that contain . Call these subspaces . Now, the key insight is that the span of is another subspace that contains ; denote it as . Then, the span is just . But since the intersection must be contained in each subspace, it is also contained in the span.

   Thus, defined by subspaces is a subset of defined by linear combinations.

Hence, we have shown that the two definitions of span are equivalent.

Linear Independence

A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the other vectors. In other words, a set of vectors is linearly independent if the only way to write the zero vector as a linear combination of the vectors is by setting all the coefficients to zero.

A set of vectors is linearly independent if the only solution to the equation:

is .

Lemma: Span and Linear Independence

There is a close relationship between the concepts of span and linear independence. In particular:

If a vector space is defined by , and given a subset of linearly independent vectors , then it follows that .

Before we prove this lemma, let's understand the intuition behind it.