3.1.3 Tensor Dual Space Addition Structure

Tensor Dual Space Addition Structure is the description of how vector-space addition acts on a dual tensor space such as V* ⊗ W*, covering the coordinatewise addition rule for elements expressed relative to a dual basis, the additive behavior of the corresponding multilinear functionals, and the way addition interacts with rank, in particular why the rank of a sum of two elements can be less than, equal to, or greater than the sum of their individual ranks, unlike scalar multiplication by a nonzero scalar, which always preserves rank exactly.

Addition of Simple and General Elements

Coordinatewise Addition

For two elements β = Σ_{i,j} b_{ij} e^i ⊗ f^j and γ = Σ_{i,j} c_{ij} e^i ⊗ f^j of V* ⊗ W*, expressed in the dual basis induced from bases {e_i} of V and {f_j} of W, their sum is:

matching the ordinary entrywise sum of the coordinate matrices (b_{ij}) and (c_{ij}). Addition of dual tensor space elements is, in coordinates, exactly matrix addition, the same relationship scalar action bears to matrix scalar multiplication.

Sum of Simple Elements Is Generally Not Simple

Although each of β and γ may be a simple element, ω ⊗ η and μ ⊗ ν respectively, their sum ω ⊗ η + μ ⊗ ν is generally not itself expressible as a single simple element unless ω and μ are scalar multiples of each other, or η and ν are; addition of simple elements is the fundamental way that non-simple, higher-rank elements of V* ⊗ W* arise.

Additivity of the Functional Interpretation

Sum of Functionals Acts Pointwise

Regarding β and γ as bilinear functionals on V × W, their sum satisfies:

for every (v, w) ∈ V × W, matching the ordinary pointwise definition of a sum of functions. This confirms that the tensor-space addition and the functional-space addition are one and the same operation viewed from two perspectives.

Zero Element as Additive Identity

The zero element of V* ⊗ W*, with all coordinates b_{ij} = 0, corresponds to the identically-zero functional and serves as the additive identity: β + 0 = β for every β, both as a coordinate statement and as a statement about the corresponding functionals.

Rank Behavior Under Addition

Rank Can Decrease

Unlike scalar multiplication by a nonzero scalar, which always leaves rank unchanged, addition can reduce rank: if γ = -β, then β + γ = 0, a rank-0 element, even though β and γ individually may have arbitrarily large rank. More generally, cancellation between terms in the decompositions of β and γ can produce a sum with strictly smaller rank than either summand.

Rank Can Increase, Bounded by the Sum

If β has rank r and γ has rank s, concatenating their minimal decompositions gives a decomposition of β + γ using at most r + s simple elements, so rank(β + γ) ≤ rank(β) + rank(γ). This bound can be strict, since a decomposition using r + s terms need not be minimal, and it can also be attained with equality, so no single, simpler formula in general replaces this inequality.

Symmetric and Alternating Parts Under Addition

Additive Decomposition Into Symmetric and Alternating Parts

For V = W, every element β ∈ V* ⊗ V* decomposes additively and uniquely as β = β_sym + β_alt, where β_sym(v, u) = (β(v, u) + β(u, v))/2 is symmetric and β_alt(v, u) = (β(v, u) - β(u, v))/2 is alternating. Addition respects this decomposition termwise: (β + γ)_sym = β_sym + γ_sym and (β + γ)_alt = β_alt + γ_alt, since the symmetrizing and alternating projections are themselves linear operations, built from addition and scalar multiplication by 1/2, and therefore commute with addition of the original elements.