3.11.1 Tensor Dual Dimension Equality

Tensor Dual Dimension Equality is the theorem that a finite-dimensional vector space V and its dual space V* always have exactly the same dimension, dim(V*) = dim(V), a foundational fact of linear algebra that can be established through several independent routes, including direct construction of a dual basis and a general formula for the dimension of spaces of linear maps. This equality is what makes V and V* interchangeable as abstract vector spaces in finite dimensions, even though, as separate spaces, they play conceptually different roles.

Proof via the Dual Basis Construction

Constructing a Basis of Matching Size

Given a basis e_1, ..., e_n of V, define e^1, ..., e^n in V* by declaring e^i(e_j) = δ^i_j and extending linearly. This produces exactly n elements of V*.

Spanning

Any f in V* satisfies f = f(e_1) e^1 + ... + f(e_n) e^n, since both sides agree when evaluated on every basis vector e_j, and a linear functional is determined entirely by its values on a basis. Hence e^1, ..., e^n spans V*.

Linear Independence

If c_1 e^1 + ... + c_n e^n = 0 as a functional, evaluating at e_j gives c_j = 0 for every j, using the delta relation to eliminate all terms except the j-th. Hence the e^i are linearly independent.

Conclusion

Since e^1, ..., e^n is both spanning and linearly independent, it is a basis of V* consisting of exactly n elements, proving dim(V*) = n = dim(V).

Proof via the Dimension Formula for Hom Spaces

General Formula for Linear Map Spaces

For finite-dimensional vector spaces V and W, the space Hom(V, W) of linear maps from V to W has dimension equal to the product of the two dimensions:

Specializing to the Dual Space

Since V* = Hom(V, F) and F itself is a one-dimensional vector space over itself, substituting W = F gives

giving the dimension equality as an immediate corollary of the general formula for linear map spaces, independent of the explicit dual basis construction.

Proof via Matrix Representation

Functionals as Row Matrices

Relative to a basis of V, every linear functional f : V -> F is represented by a 1 x n matrix, since it maps n-dimensional column vectors to scalars. The vector space of all 1 x n matrices over F has dimension n, matching the number of independent entries in such a matrix, and this vector space of matrices is precisely a coordinate realization of V*.

Consistency Across Methods

All three approaches, the direct dual basis construction, the general Hom dimension formula, and the matrix representation argument, converge on the identical conclusion, reinforcing that the dimension equality is a robust structural fact rather than an artifact of any one particular method of proof.

Significance of the Equality

Enabling a Choice of Isomorphism

Because V and V* share the same dimension, they are isomorphic as abstract vector spaces, though, as previously discussed, no particular isomorphism between them is canonical without further structure such as a fixed basis or inner product.

Symmetry in Tensor Space Dimension Formulas

The dimension equality also ensures that the general tensor space T^p_q(V), built from p copies of V and q copies of V*, has dimension n^{p+q} regardless of how the p + q factors are distributed between V and V*, since every factor contributes the same dimension n to the total count of basis tensor products.

Diagrammatic Summary

The diagram shows three independent lines of reasoning all converging on the same dimension equality between a finite-dimensional space and its dual.