2.13.1 Tensor Scalar Multiplication Closure

Tensor Scalar Multiplication Closure is the guarantee that scaling any vector from a vector space by any scalar from the underlying field always produces another vector that remains within that same space, so the scaling operation never produces a result outside the space it is defined on. This closure, together with closure under addition, is one of the two operation-based conditions required for a subset of a vector space to qualify as a subspace.

Formal Statement

Closure as Part of the Vector Space Definition

For any vector drawn from the vector space and any scalar drawn from the field, the scaled vector is guaranteed to remain within that same vector space.

Closure Restricted to a Subspace

When testing whether a subset of a vector space is a subspace, closure under scalar multiplication requires that scaling any element of the subset by any field element produces a result still inside the subset, not merely inside the larger ambient space.

Role in the Standard Subspace Test

One of the Three Standard Criteria

Closure under scalar multiplication, together with zero vector membership and closure under addition, forms the standard three-part test for whether a subset qualifies as a subspace of the ambient vector space.

Deriving Zero Vector Membership From This Closure

Closure under scalar multiplication alone implies that the zero vector belongs to any nonempty subset satisfying it, since scaling any element of the subset by the scalar zero produces the zero vector, which by closure must remain in the subset.

Closure Behavior in Coordinates

Entrywise Closure of Coordinate Vectors

Because scalar multiplication corresponds to multiplying every entry of a coordinate tuple by the same scalar, closure under scalar multiplication guarantees that scaling the coordinate vector of an element of the space produces a coordinate tuple that itself corresponds to a valid vector in the space.

Uniform Application Across All Positions

This closure holds uniformly across every coordinate position, since the scalar multiplies each entry independently, so no special exception arises for any particular component of the coordinate tuple.

Role in Tensor Construction

Ensuring Scaled Tensors Remain Valid

Closure under scalar multiplication within each factor vector space supports the broader fact that scaling a tensor built from those factors by a field element produces another valid tensor within the same tensor space, preserving the algebraic structure at the tensor level.

Interaction With Span Closure

Scalar multiplication closure is directly connected to span closure, since the closure of a span under scalar multiplication, established separately for spans, is a specific instance of this more general closure property applied to the subset formed by the span.

Summary of Key Properties

No Escape From the Vector Space Under Scaling

Tensor Scalar Multiplication Closure guarantees that scaling any vector by any field element always produces a result that remains within the same vector space or subspace under consideration.

Complementary to Addition Closure

Together with closure under addition, this property completes the pair of operation-based closure conditions that, alongside zero vector membership, define what it means for a subset to behave as a subspace.