2.13 Tensor Scalar Multiplication Operation

Tensor Scalar Multiplication Operation is the second fundamental operation defined on a vector space used for tensor construction, taking a scalar from the underlying field and a vector from the space and producing a new vector obtained by scaling, with this operation required to satisfy a specific set of algebraic laws alongside vector addition. Scalar multiplication is what introduces the field's arithmetic into the vector space, allowing vectors to be stretched, shrunk, or reversed in a manner compatible with the field's own operations.

Formal Statement

Definition as an External Binary Operation

Scalar multiplication takes a scalar from the field and a vector from the space and produces another vector from the same space, combining an element from outside the vector space with an element inside it.

Required Algebraic Laws

Scalar multiplication must respect multiplication in the field, distribute over vector addition, distribute over field addition, and leave vectors unchanged when scaled by the field's multiplicative identity.

Distributive Interactions

Distribution Over Vector Addition

Scaling the sum of two vectors by a single scalar produces the same result as scaling each vector separately and then adding the results.

Distribution Over Field Addition

Scaling a single vector by the sum of two scalars produces the same result as scaling the vector by each scalar separately and then adding the resulting vectors.

Behavior in Coordinates

Component Form of Scalar Multiplication

Relative to a fixed basis, scaling a vector by a scalar corresponds to multiplying every entry of its coordinate tuple by that same scalar, so the operation acts uniformly across all coordinate positions.

Preservation of Coordinate Length

Since scalar multiplication does not change which basis is used or how many basis vectors exist, it never alters the length of a coordinate tuple, only the numeric values held within it.

Role in Tensor Construction

Providing the Field Action Needed for Tensor Coefficients

Scalar multiplication supplies the mechanism by which tensors, built from vectors that support this operation, can themselves be scaled by field elements, extending the field's action from individual vectors up to the level of full tensors.

Interaction With Bilinearity of the Tensor Product

The tensor product operation respects scalar multiplication in each of its factor slots, meaning a scalar can be moved freely between a factor and the tensor product as a whole, a property that depends directly on the compatibility laws satisfied by scalar multiplication.

Summary of Key Properties

Field-Compatible Scaling of Vectors

Tensor Scalar Multiplication Operation establishes a scaling action of the field on the vector space that is fully compatible with both the field's own arithmetic and the vector space's addition operation.

Second Pillar of Vector Space Structure

Alongside vector addition, scalar multiplication forms the second essential operation whose laws define what it means for a set equipped with these two operations to be a vector space at all.