4.2.3 Tensor Bilinear Slotwise Linearity

Tensor Bilinear Slotwise Linearity is the property of a rank-2 tensor's associated bilinear map whereby linearity holds independently in each of its two argument slots, meaning the map behaves linearly when one argument is held fixed and the other varies, without requiring any joint linearity across both arguments simultaneously.

Precise Formulation

The Two Separate Linearity Conditions

For a bilinear map B: V × V → ℝ, slotwise linearity consists of two independent conditions. Linearity in the first slot requires

for every fixed v, and linearity in the second slot requires

for every fixed u. Neither condition alone implies the other, and both together are what the term bilinear, applied to a rank-2 tensor's associated map, precisely means.

Distinction from Joint Linearity

Slotwise linearity is strictly weaker than treating B as a single linear map on the direct sum V ⊕ V. In particular, B is generally not additive across the two slots together, meaning B(u + v, u + v) does not decompose as a simple sum unless the cross terms B(u, v) and B(v, u) are accounted for separately,

which already shows B is quadratic, not linear, when both arguments vary together.

Coordinate Consequences

Component Extraction via Slotwise Linearity

Slotwise linearity is exactly what permits a bilinear tensor to be fully characterized by its values on basis pairs. Writing u = u^i e_i and v = v^j e_j and applying linearity first in the first slot and then in the second,

Each factor of u^i and v^j is pulled outside the map by applying slotwise linearity to one slot at a time, and this two-step extraction is only valid because each slot is linear on its own; no assumption of joint linearity is used or required.

The n by n Matrix as a Direct Consequence

Because slotwise linearity reduces the evaluation of B on arbitrary vectors to a double sum over the n^2 basis pair values B(e_i, e_j) = B_ij, the bilinear map is completely encoded in an n × n matrix, with the bilinear evaluation reproduced by the quadratic-looking but structurally bilinear expression u^T B v.

Slotwise Linearity and Tensor Type

Consistency Across Mixed Slot Types

Slotwise linearity applies identically regardless of whether a given slot accepts a vector or a covector, since linearity in a slot is a statement about the additive and scalar structure of that slot's input space, not about which space it is. A type (1, 1) tensor T(ω, v) is therefore linear in ω over V* and linear in v over V, independently, exactly matching the pattern for a type (0, 2) or type (2, 0) tensor.

Failure of Slotwise Linearity Under Nonlinear Substitution

If a nonlinear function is substituted into a slot, for example replacing v by a function of another variable that is not itself linear, the resulting expression is no longer bilinear in the original sense, even though B itself remains bilinear as a map on its own domain. This distinction matters when composing bilinear tensors with nonlinear coordinate changes, since slotwise linearity is a property of B as an abstract map, not automatically inherited by every composite expression built from it.

Role in Pullback of Bilinear Tensors

Preservation of Slotwise Linearity Under Pullback

Given a linear map f: V → W and a bilinear form B on W, the pullback (f*B)(u, v) = B(f(u), f(v)) is itself bilinear on V, because f is linear in each argument and B is slotwise linear in each of its arguments, so the composition remains linear in u and in v separately. This preservation is essential: pullback of a bilinear tensor along a linear map would fail to produce another tensor at all if slotwise linearity were not maintained through the composition.