3.3.5 Tensor Linear Functional Kernel Structure

Tensor Linear Functional Kernel Structure is the detailed study of the kernel of a linear functional as a subspace in its own right, extending beyond the basic codimension-one fact to cover how kernels of several functionals intersect, how linear (in)dependence among a set of functionals is detected through their common kernel, and how the kernel of a functional interacts with the kernel of related functionals obtained by scaling or restriction.

The Kernel as a Subspace

Closure Properties

For ω ∈ V*, ker(ω) = {v ∈ V : ω(v) = 0} is a subspace: it contains 0, since ω(0) = 0 by homogeneity with c = 0; it is closed under addition, since ω(v_a) = ω(v_b) = 0 gives ω(v_a + v_b) = ω(v_a) + ω(v_b) = 0; and it is closed under scalar multiplication, since ω(v) = 0 gives ω(cv) = c ω(v) = 0. These closure properties follow directly from the linearity of ω, matching the general fact that the preimage of the zero subspace under any linear map is a subspace of the domain.

Intersection of Multiple Kernels

Kernel of a Set of Functionals

For functionals ω_1, ..., ω_k ∈ V*, the joint kernel ∩_i ker(ω_i), the set of vectors annihilated by every ω_i simultaneously, is itself a subspace, being an intersection of subspaces. Equivalently, this joint kernel is the kernel of the single linear map Φ : V → F^k sending v to (ω_1(v), ..., ω_k(v)), so the codimension of the joint kernel is at most k, dim(V) - dim(∩_i ker(ω_i)) ≤ k, with equality exactly when ω_1, ..., ω_k are linearly independent as elements of V*.

Linear Independence and Codimension

If ω_1, ..., ω_k are linearly independent in V*, the map Φ above is surjective onto F^k, so by rank-nullity dim(∩_i ker(ω_i)) = dim(V) - k exactly; each additional independent functional in the collection reduces the joint kernel's dimension by exactly one, while a dependent functional, expressible as a combination of the others, contributes no further reduction beyond what the independent ones already impose.

Common Kernel and Linear Dependence

Redundant Functionals Share Larger Kernels

If ω_2 = c ω_1 for a nonzero scalar c, then ker(ω_1) = ker(ω_2) exactly, since ω_1(v) = 0 if and only if c ω_1(v) = 0. More generally, if ω_k is a linear combination Σ_{i<k} c_i ω_i of the others, then ∩_{i<k} ker(ω_i) ⊆ ker(ω_k), since any vector annihilated by every ω_i for i < k is automatically annihilated by any linear combination of them; a dependent functional's kernel already contains the joint kernel of the functionals it depends on.

Detecting Independence via Kernel Dimension

Because independent functionals strictly shrink the joint kernel while dependent ones do not, the sequence of joint kernel dimensions dim(V) ≥ dim(ker(ω_1)) ≥ dim(ker(ω_1) ∩ ker(ω_2)) ≥ ⋯ strictly decreases at each step exactly when the newly added functional is independent of the previous ones, and stays constant exactly when it is dependent; tracking this sequence is a direct, kernel-based method for testing linear independence of a set of functionals.

Kernel Under Scalar Multiples and Sums

Scalar Multiples Preserve the Kernel

For nonzero c, ker(cω) = ker(ω), since cω(v) = 0 if and only if ω(v) = 0 when c ≠ 0; scaling a functional by a nonzero constant never changes which vectors it annihilates, consistent with the earlier observation that two functionals sharing a kernel differ by exactly such a scalar multiple.

Sums Generally Have a Different, Larger or Smaller Kernel

By contrast, ker(ω_1 + ω_2) is generally neither ker(ω_1) nor ker(ω_2) nor their intersection; a vector v lies in ker(ω_1 + ω_2) whenever ω_1(v) = -ω_2(v), a condition satisfied by vectors in ker(ω_1) ∩ ker(ω_2) but potentially by many other vectors as well, so ker(ω_1) ∩ ker(ω_2) ⊆ ker(ω_1 + ω_2) holds in general, but this inclusion is typically strict.

Kernel Structure and the Annihilator

The Kernel as a Special Case of the Annihilator

ker(ω) = ({ω})°, the annihilator of the one-element set (equivalently, the one-dimensional subspace spanned by ω, when ω ≠ 0) inside V, treating V and V** as identified via the canonical double-dual isomorphism; the joint kernel of several functionals ω_1, ..., ω_k is likewise the annihilator, under this identification, of the subspace of V* that they span, connecting kernel structure directly to the annihilator construction already established for subspaces of V and showing that the two notions, kernel of functionals and annihilator of subspaces, are dual expressions of the same underlying pairing.