2.5 Tensor Vector Space Basis Structure

Tensor Vector Space Basis Structure is the construction, from a basis of the underlying vector space, of a basis for the space of tensors of a fixed type, together with the proof that the resulting set of tensor products is both spanning and linearly independent. It is the structural fact that reduces the study of an abstract, potentially high-dimensional tensor space to a finite, explicitly enumerable list of elementary building blocks.

Construction of the Induced Basis

Setting

Let $$V$$ be a vector space over a field $$F$$ with basis $$e_1,\cdots ,e_n$$, and let $$e^1,\cdots ,e^n$$ be the corresponding dual basis of $$V^{*}$$, satisfying $$e^i\left(e_j\right)={\delta }_j^i$$.

The Candidate Basis Set

For a tensor space $$T_s^r\left(V\right)$$ of type $$\left(r,s\right)$$, the induced basis consists of every tensor product formed by choosing one basis vector for each contravariant slot and one dual basis covector for each covariant slot:

ranging over all index tuples $$i_1,\cdots ,i_r,j_1,\cdots ,j_s$$ with each index between $$1$$ and $$n$$.

Spanning Property

Every Tensor Is a Combination of Basis Tensors

Given any $$T\in T_s^r\left(V\right)$$, multilinearity guarantees the expansion

with summation over repeated indices, where the coefficients are the components of $$T$$ in this basis. Since this expansion holds for every element of the space, the candidate set spans $$T_s^r\left(V\right)$$.

Linear Independence

The Independence Argument

Suppose a linear combination of the candidate basis tensors vanishes:

Evaluating both sides on the input tuple $$\left(e^{k_1},,,\cdots ,,,e_{l_1},,,\cdots \right)$$ and using the duality relation $$e^i\left(e_j\right)={\delta }_j^i$$, every term vanishes except the one matching the chosen indices exactly, forcing

for every index combination. Since this holds for all choices of $$k_1,\cdots ,l_1,\cdots$$, every coefficient in the original combination is zero, establishing linear independence.

Dimension of the Tensor Space

Counting the Basis Elements

The number of index tuples $$\left(i_1,,,\cdots ,,,i_r,,,j_1,,,\cdots ,,,j_s\right)$$, each ranging independently over $$n$$ values, is $$n^{r+s}$$. Since the induced set both spans and is linearly independent, it is a basis, and

where $$n=\text{dim}V$$.

Uniqueness of Coordinates Relative to the Basis

Well-Defined Coefficients

Since the induced set is a basis, the coefficients appearing in the expansion of any tensor $$T$$ are uniquely determined; no other assignment of coefficients to the same basis tensors produces the same element. These uniquely determined coefficients are exactly the components of $$T$$ introduced elsewhere in the coordinate description of tensors.

Special Cases

Vectors and Covectors

For type $$\left(1,0\right)$$, the induced basis reduces to the original basis $$e_1,\cdots ,e_n$$ of $$V$$; for type $$\left(0,1\right)$$, it reduces to the dual basis $$e^1,\cdots ,e^n$$.

Matrices

For type $$\left(1,1\right)$$, the induced basis consists of the $$n^2$$ elementary tensors $$e_i\otimes e^j$$, which correspond under the standard identification with linear endomorphisms of $$V$$ to the elementary matrices having a single entry of $$1$$ and all other entries $$0$$.

Dependence on the Choice of Underlying Basis

Different Bases Give Different Basis Tensors

A different choice of basis for $$V$$ induces a different basis for $$T_s^r\left(V\right)$$, and the two induced bases are related by the tensor transformation law applied to each factor. The dimension $$n^{r+s}$$ is unchanged by this choice, since it depends only on $$n$$, the invariant dimension of $$V$$, and on the fixed type $$\left(r,s\right)$$.

Basis Structure Underlies Coordinate Computation

This basis structure is precisely what licenses the coordinate description of tensors: every computational technique that represents a tensor by an indexed array of components implicitly relies on the fact that the induced tensor products of basis vectors and dual basis covectors form a genuine basis, spanning the space without redundancy.