Question

Given a linear space \(\mathrm{L}\), a set \(\\{\mathrm{x}\), ) of linearly independent elements of \(\mathrm{L}\) is said to be a Hamel basis (in \(\mathrm{L}\) ) if the linear subspace generated by \(\\{\mathrm{x}\), ) coincides with L. Prove that a) Every linear space has a Hamel basis; b) If \(\\{x\), ) is a Hamel basis in \(\mathrm{L}\), then every vector \(\mathrm{x} E \mathrm{~L}\) has a unique representation as a finite linear combination of vectors from the set \(\left\\{x_{\alpha}\right\\}\) c) Any two Hamel bases in a linear space \(\mathrm{L}\) have the same power (cardinal number), called the algebraic dimension of \(\mathrm{L}\); d) Two linear spaces are isomorphic if and only if, they have the same algebraic dimension.

Short answer

Every linear space has a Hamel basis, and each vector in \( L \) has a unique finite representation with this basis. All Hamel bases have the same cardinality, defining \( L \)'s algebraic dimension. Two spaces are isomorphic if and only if they share this dimension.

Step-by-step solution

Understanding Hamel Basis

A Hamel basis for a linear space \( L \) is a set of linearly independent elements whose linear combinations can generate every element of \( L \). To comprehensively understand a linear space, it's crucial to find such a basis.

Proving Existence of Hamel Basis

To prove every linear space \( L \) has a Hamel basis, assume \( L \) is non-trivial. Use Zorn's Lemma, which requires every chain of linearly independent subsets in \( L \) has an upper bound, meaning there's a maximal set of linearly independent vectors which covers \( L \). This maximal set forms a Hamel basis.

Uniqueness of Representation

Given a Hamel basis \( \{x_\alpha\} \), any vector \( x \in L \) can be uniquely expressed as \( x = a_1 x_{\alpha_1} + a_2 x_{\alpha_2} + ... + a_n x_{\alpha_n} \), where \( a_i \) are scalars. The uniqueness follows from the linear independence of \( \{x_\alpha\} \). If there existed two different combinations producing \( x \), then their difference would violate independence.

Cardinality Equality of Hamel Bases

Suppose \( L \) has two distinct Hamel bases \( \{x_\alpha\} \) and \( \{y_\beta\} \). Use bijections: each basis vector from \( \{x_\alpha\} \) is a finite linear combination of \( \{y_\beta\} \), and vice versa. This implies both sets have the same cardinal number.

Isomorphism and Algebraic Dimension

Two linear spaces \( L_1 \) and \( L_2 \) are isomorphic if one can map every basis vector of one space to a basis vector of the other. If their algebraic dimensions (sizes of their bases) are the same, there's a bijection between the bases, establishing an isomorphism. Conversely, being isomorphic ensures matching dimensions as bases can be relabeled to match.

Key concepts

Linear Independence

Linear independence is a foundational concept in linear algebra. It refers to a set of vectors in a vector space that do not linearly "depend" on each other. In simpler terms, none of the vectors in the set can be written as a linear combination of the others. This independence is crucial because it ensures that each vector contributes something unique to the space. If we think of a vector as a direction or a way of moving in space, linearly independent vectors each offer a new direction that isn't redundant.

Why is this important? Knowing that a set of vectors is linearly independent tells us that they can form a basis for the space. In a more practical sense:

• They provide a way to express all other vectors in the space as unique combinations of them.
• No vector in this set can be neglected while maintaining the ability to form the space.

Understanding linear independence helps us see why certain sets of vectors are chosen as bases, like the Hamel basis. It simplifies calculations and descriptions of complex spaces.

Zorn's Lemma

Zorn's Lemma is an essential tool in modern mathematics used to prove the existence of certain mathematical objects when direct construction is difficult or impossible. In the context of proving the existence of a Hamel basis, Zorn's Lemma helps us overcome the challenge of building a maximal set of linearly independent vectors.

Here’s how it works:

• Suppose we have a chain (or sequence) of linearly independent sets in the linear space.
• Zorn's Lemma states that if each chain has an upper bound (a maximal element), then there is a maximal set of linearly independent vectors.
• This maximal set corresponds to the Hamel basis we seek for the linear space.

Zorn's Lemma can be thought of as a "bridging" principle that allows us to go from potentially smaller subsets to a complete set (in this case, the Hamel basis) that fully spans the linear space, ensuring we capture the entire structural essence of our space.

Algebraic Dimension

The algebraic dimension of a linear space is the cardinal number representing the size of any Hamel basis for that space. This concept is fundamental because it provides a rigorous way to compare different linear spaces by looking at the size of their bases rather than their apparent dimensions.

Key things to remember about algebraic dimension:

• If two different Hamel bases exist in the same linear space, they will have the same cardinal number. This is assured by the fact that any basis can be expressed as a finite combination of another basis, following a bijective relationship.
• This means that the algebraic dimension is a true invariant, a characteristic intrinsic to the structure of the space itself.

Understanding the algebraic dimension is crucial when studying linear spaces because it allows us to determine their size and capacity in a consistent and standardized manner. This, in turn, lays the groundwork for comprehending more advanced topics like isomorphisms.

Isomorphism of Linear Spaces

Isomorphism is a concept that finds its place prominently in the study of linear spaces. Two linear spaces are said to be isomorphic if there exists a bijective linear map between them. This map ensures that the spaces, while possibly different in appearance, have the same structure and properties.

Why does isomorphism matter? Because it indicates a deep level of equivalency between spaces:

• Isomorphic linear spaces allow for "relabeling" of basis vectors between them, given their bases have the same algebraic dimension.
• This relabeling aligns the structure of one space perfectly with the other, making them identical in terms of operations and transformations.

This notion of isomorphism answers questions about when two spaces can be treated as the same in the context of linear algebra. If they have equal algebraic dimensions, they can be considered structurally identical. Understanding this can simplify problems and solutions by recognizing underlying similarities in different spaces.