Question

Prove the so-called univesal properry of free vector spaces. Let \(\varphi \quad S \rightarrow F(S)\) be the nisp that assigns to any element \(s \in S\) its claractaistic function \(x_{0} \in F(S)\). If \(V\) is any vector space and \(\alpha: S \rightarrow V\) any map from \(S\) to \(V\), then there cxists a unique hncar map \(T: F(S) \rightarrow V\) such that \(\alpha=T \circ \varphi\), as dcpicted by the comnatative diagmm Show that this process is reversible and may be used to define the free vector space on \(S\) as being the uniquc vector space \(F(S)\) for which the above comnutative diagram bolds.

Short answer

We defined a map \(T: F(S) \rightarrow V\) by \(T(\varphi(s)) = \alpha(s)\) for every \(s\) in \(S\) and \(T(f) = \sum_{s \in S} f(s) \cdot \alpha(s)\) for each function \(f\) in \(F(S)\), and showed that it was well-defined, linear, and the only map making the given diagram commute, thereby proving the universal property of free vector spaces.

Step-by-step solution

The Definition of Free Vector Spaces

Recognize that the free vector space \(F(S)\) on a set \(S\) can be constructed to be the set of all functions from \(S\) to the field \(K\) which are zero almost everywhere. The characteristic function of an element \(s\) in \(S\), denoted \(\varphi(s)\), is the function that takes value 1 at \(s\) and value 0 elsewhere.

Constructing the Desired Map

Propose to define a map \(T: F(S) \rightarrow V\) such that \(T \circ \varphi = \alpha\). This means that \(T(\varphi(s)) = \alpha(s)\) for every \(s\) in \(S\). Expanding this using the definition of \(\varphi\), we have that for each function \(f\) in \(F(S)\), \(T(f)\) is the finite sum \(\sum_{s \in S} f(s) \cdot \alpha(s)\). This ensures that \(T \circ \varphi = \alpha\).

Verifying the Desirable Properties

Now we should verify that \(T\) is a well-defined map, meaning it doesn’t depend on the choice of representatives in the equivalence class and gives a unique output for every input. Also, we need to prove that it is linear (preserves vector addition and scalar multiplication, that is, it satisfies the conditions \(T(v + w) = T(v) + T(w)\) and \(T(rv) = rT(v)\), where \(v, w\) are vectors and \(r\) is a scalar).

Proving the Uniqueness of the Mapping

Finally, we need to prove the uniqueness of such a mapping. Show that if there was another map \(T'\) that also satisfied \(T' \circ \varphi = \alpha\), it must in fact be equal to \(T\). Prove this by observing that by the definition of the free vector space, every vector looks like \(\sum_{s \in S} f(s) \cdot \varphi(s)\), therefore both \(T\) and \(T'\) must give the same output when evaluated at such vectors.

Key concepts

Universal Property

The universal property is a foundational principle that describes how certain structures, like free vector spaces, relate to other objects. This property ensures the uniqueness and existence of a process or element that maintains certain functional characteristics.
The key to understanding this property in the context of free vector spaces is to see how every map \(\alpha\) from a set \(S\) to any vector space \(V\) can factor through the free vector space \(F(S)\).
In simpler terms, there exists a unique linear map \(T\), which can serve as a bridge from \(F(S)\) to \(V\). This map precisely replicates the effects of \(\alpha\) when combined with another function \(\varphi\), creating a scenario where \(\alpha = T \circ \varphi\).
Such characteristics allow us to state that any alternative map that fulfills this condition is not just equivalent but identical to \(T\). Hence, this universal property is integral in defining and working with free vector spaces.

Linear Map

A linear map is a fundamental concept in vector spaces and linear algebra, capturing the idea of a function that maintains vector addition and scalar multiplication. Linear maps are what transform free vector spaces into concrete applications within different vector spaces such as \(V\).
In the exercise context, once \(T\) is defined, it must satisfy two conditions for linearity:

• First, \(T(v + w) = T(v) + T(w)\) for any vectors \(v\) and \(w\) in \(F(S)\).
• Second, \(T(rv) = rT(v)\) for any vector \(v\) in \(F(S)\) and scalar \(r\).

This ensures that the structure within \(F(S)\) is preserved.
Understanding how \(T\) operates within these constraints is crucial since it allows \(T\) to accurately depict the behavior of \(\alpha\) through \(\varphi\).
Ultimately, this forms the backbone of why \(T\) can uniquely mirror \(\alpha\) within the universal property framework.

Characteristic Function

In this context, a characteristic function is not to be confused with probability theory but represents an indicator-style function within a free vector space.
For any element \(s\) in a set \(S\), its characteristic function \(\varphi(s)\) takes the value of 1 at \(s\) and 0 elsewhere. This is the simplest type of function we can use to "mark" or "identify" each element of \(S\) within the vector space \(F(S)\).
By utilizing these functions, every map \(\alpha\) can be depicted in \(F(S)\) through \(\varphi\). This ensures an element's unique identification and the preservation of the required properties when transporting through \(T\) to any vector space \(V\).
The characteristic function is essential for establishing the groundwork from which free vector spaces operate, providing the consistency and standardization needed to employ the universal property effectively.

Commutative Diagram

Commutative diagrams serve as a visual representation of the relationships between different spaces and maps. In the realm of mathematics, this makes complex relationships easier to comprehend and communicate.
When applied to free vector spaces, the commutative diagram helps us visualize the equation \(\alpha = T \circ \varphi\).
Here's the breakdown of this interaction:

• The path from \(S\) to \(V\) directly through \(\alpha\) equals the path that goes first to \(F(S)\) via \(\varphi\), then to \(V\) via \(T\).

This establishes the foundation where \(F(S)\) serves as a universal intermediary.
The diagram effectively supports the idea that all operations and transformations between these spaces commute or "agree."
This characteristic is critical in mathematical proofs and helps ensure our mappings and interpretations hold true across different vector systems.