Can Column Vectors Represent All Possible Vectors in a Vector Space and How to Deduce From Axioms?

The question of whether column vectors can represent all possible vectors in a vector space is a fundamental one within linear algebra. This concept is particularly crucial for understanding the nature of vector spaces and the effective representation of their elements.

Different Representations of Vectors in a Vector Space

In a Euclidean space like Fn, where F is a numerical field (such as the real numbers R or the complex numbers C), column vectors are a convenient way to represent vectors. An element in such a space is an ordered n-tuple X x1x2...xn, which can be written as a column vector X [x1 x2 ... xn]sup>T. Alternatively, this element can be represented as a row vector or an n-tuple, but the column vector form is often preferred for its simplicity and convenience in computing linear combinations and matrix operations.

Operations and Representations

The operations involved in defining a vector space, such as vector addition and scalar multiplication, are rigorously defined. For example, given two vectors X [x1 x2 ... xn]sup>T and Y [y1 y2 ... yn]sup>T, their sum is defined as:

X Y [x1 y1 x2 y2 ... xn yn]sup>T

And for a scalar λ, the scalar multiplication of a vector X is defined as:

λX [λx1 λx2 ... λxn]sup>T

Abstract vs. Specific Spaces

In abstract general vector spaces, vectors cannot be represented solely as column vectors unless the space is specifically Fn, Rn, or Cn. These specific spaces are examples where vectors can be represented as column vectors due to their closed nature under vector addition and scalar multiplication, which are axioms L1 and L6 of a vector space. However, not all vector spaces satisfy these conditions, and thus their vectors may not be representable solely as column vectors.

Deduction from Axioms

While the axioms of a vector space (L1...L10) do not explicitly deduce the representation of vectors as column vectors, they do provide the necessary framework. Axioms L1 and L6, specifically, are crucial because they define the operations of vector addition and scalar multiplication, which are essential for the operations in the Fn space.

For a subset W of a vector space V, to determine whether it is a subspace, only axioms L1 and L6 need to be checked since the other axioms (such as associativity of the sum and the existence of the zero vector) are implied by the inclusion in the parent space. This simplifies the verification process, making axioms L1 and L6 particularly important in this context.

Linear Dependence and Independence

Column vectors play a significant role in studying linear dependence and independence of a set of vectors, which is essential for solving linear systems. A linear system AX b can be studied through the matrix representation of the vectors and the coefficients. This system can be solved to establish the rank of the matrix, a crucial parameter in understanding the dependency relations within the system.

Conclusion

The use of column vectors for representing vectors in vector spaces, particularly in Fn, Rn, or Cn, is a practical and efficient approach, though not every vector space can be represented in this manner. The axioms defining vector spaces provide the necessary structure, with axioms L1 and L6 being particularly important for operations and representations in the context of these specific spaces.