2.10.4 Tensor Coordinate Vector Column Form

Tensor Coordinate Vector Column Form is the convention of writing a coordinate vector as a vertical arrangement of its components, stacked from top to bottom in the order determined by the underlying ordered basis, so that the resulting column can be manipulated directly using matrix notation and matrix operations. Column form is the standard presentation used whenever coordinate vectors are combined with linear maps expressed as matrices.

Formal Statement

Vertical Arrangement of Components

A coordinate vector with components c1 through cn is written in column form as a single-column array with n rows, each row holding one component in the order fixed by the ordered basis.

Row Position Matches Basis Position

The entry appearing in row i of the column exactly matches the i-th component of the coordinate vector, which in turn is the coefficient of the i-th basis vector in the fixed ordered basis.

Compatibility With Matrix Operations

Matrix-Vector Multiplication

Column form allows a linear map represented by a matrix to act on a coordinate vector through ordinary matrix-vector multiplication, producing the coordinate vector of the image under that linear map.

Entrywise Addition as Column Addition

Adding two coordinate vectors in column form corresponds to adding the columns entrywise, matching row against row, which mirrors the coefficientwise addition already established for coordinate vectors in general.

Alternative to Row Form

Contrast With Row Vectors

Column form is distinguished from row form, in which the same components are arranged horizontally, and the choice between the two affects how multiplication with matrices is written, though both forms carry the identical component data.

Convention Rather Than Structural Necessity

The preference for column form in most tensor and linear algebra contexts is a notational convention chosen for compatibility with the standard convention of applying matrices on the left of vectors, rather than a structural requirement of the vector space itself.

Role in Tensor Construction

Stacking Coordinate Data for Tensor Components

Column form provides a convenient way to organize the coordinate data of factor vectors before they are combined into the multi-indexed component structure of a tensor, since each factor's column can be referenced independently by its row positions.

Interface With Coordinate Vector Ordering

Column form directly depends on coordinate vector ordering, since the vertical sequence of entries in the column is meaningless without the fixed order of the underlying basis that assigns each row its identity.

Summary of Key Properties

Standard Presentation for Computation

Tensor Coordinate Vector Column Form gives coordinate vectors a standard vertical presentation that integrates directly with matrix-based linear algebra computations.

Preserves All Coordinate Vector Properties

Writing a coordinate vector in column form changes only its visual arrangement, not its mathematical content, so all properties established for coordinate vectors in general, including basis dependence and dimension-fixed length, carry over unchanged.