1.2.9 Basis Definition

Basis Definition is the characterization of a basis of a vector space as a set of vectors that is linearly independent and spans the entire space, meaning that every vector in the space can be written as a unique linear combination of the basis vectors. It provides the mechanism by which an abstract vector space is given a concrete system of coordinates, and it is the concept relative to which the numerical components of vectors, covectors, and tensors are ultimately expressed.

The Two Defining Conditions

A set of vectors qualifies as a basis for a vector space if it satisfies two conditions simultaneously. First, the set must be linearly independent, meaning that no vector in the set can be written as a linear combination of the others, or equivalently, that the only linear combination of the set's vectors equaling the zero vector is the one in which every coefficient is zero. Second, the set must span the space, meaning that every vector in the space can be obtained as some linear combination of the vectors in the set.

The expression above states that a vector can be written as a linear combination of basis vectors, with coefficients drawn from the underlying scalar field. When the set of vectors used in this expansion forms a basis, this representation exists and is unique for every vector in the space.

Uniqueness of Representation

The combination of linear independence and spanning guarantees that every vector in the space has exactly one representation as a linear combination of the basis vectors: spanning ensures that at least one such representation exists, while linear independence ensures that no vector can be represented in two different ways, since the difference of two such representations would otherwise give a nontrivial linear combination equaling zero, contradicting independence. This uniqueness is what allows the coefficients in the expansion, called the coordinates of the vector relative to the chosen basis, to be treated as a well-defined numerical description of that vector.

Existence and Cardinality of a Basis

Every vector space, under standard assumptions, possesses at least one basis, though the specific vectors that make up a basis are never unique: infinitely many different bases exist for any vector space of dimension greater than zero. What remains invariant across every possible choice of basis is the number of vectors the basis contains, called the dimension of the vector space. This invariance is a foundational theorem of linear algebra, guaranteeing that dimension is a well-defined property of the vector space itself, rather than an artifact of a particular basis chosen to describe it.

Change of Basis

Because a vector space admits many different bases, the same vector will, in general, have different coordinates depending on which basis is used to express it. A change of basis is described by a set of coefficients specifying how the vectors of a new basis are expressed in terms of an old basis, and this same set of coefficients determines precisely how the coordinates of any vector must be recalculated when passing from one basis to the other. This relationship between changes of basis and changes of coordinates is the direct source of the tensor transformation law: a tensor's components change under a change of basis according to a rule built from exactly these basis-change coefficients, applied once for each index the tensor carries.

Role of the Basis in Tensor Algebra

The basis is the single most important tool for converting the abstract, coordinate-free objects of tensor algebra — vectors, covectors, multilinear maps, and tensors of every rank — into explicit numerical arrays that can be used in direct calculation. Once a basis for a vector space is fixed, a corresponding dual basis for its dual space can be constructed, and together these allow every tensor built from the space and its dual to be expressed as an indexed array of components. Every subsequent discussion of tensor components, tensor transformation laws, and coordinate-based tensor operations depends on this prior choice of basis, making the basis definition an indispensable link between the abstract and computational sides of tensor algebra.