Where does basis collide?

In linear algebra, a basis of a vector space is a set of vectors that is linearly independent and also spans the space, so every element can be written in exactly one way as a weighted sum of those vectors. Grassmann's 1844 Die lineale Ausdehnungslehre introduced linear independence and the idea that a space is generated by a set of independent units, and Peano's 1888 Calcolo geometrico gave the axioms of a vector space in which such an independent generating set fixes every coordinate. The number of vectors in a basis is the dimension of the space. Here the referent is an abstract collection of vectors that assigns coordinates to a space, not a group of atoms placed at a lattice point or a difference between two market prices.[1][2]

Unit

One linearly independent spanning set; its count of vectors is the dimension

Recovery means

Not applicable

Measured by

Confirmed by checking that the vectors are linearly independent and span, for example by row-reducing their coordinate matrix

In crystallography, the basis (also called the motif) is the single atom or fixed group of atoms attached to every point of a crystal's lattice, so that copying the basis onto each lattice point reproduces the whole structure. Bravais's 1850 Mémoire sur les systèmes formés par des points distribués régulièrement established the regular point lattices on which crystals are built, and the description of a structure as a lattice together with a basis is set out in Kittel's Introduction to Solid State Physics. The lattice supplies only the repetition, while the basis supplies the actual atomic content. This referent is a physical arrangement of atoms in a solid, unrelated to a spanning set of vectors or a gap between two market prices.[3][4]

Unit

One atom or fixed atom group repeated at every lattice point

Recovery means

Not applicable

Measured by

Determined by X-ray, neutron, or electron diffraction, which resolves the positions of the atoms in the repeating group

In commodity finance, the basis is the difference between the current cash price of a good and the price of a futures contract on it, and it tells a hedger how well a futures position will track the cash market. Working's 1949 The theory of price of storage gave the basis its formal treatment, showing that the cash-to-futures gap reflects the cost of holding a good over time, though the term itself was already trade usage in grain markets before then. As the futures contract nears delivery the basis tends toward zero. This referent is a money quantity that moves with storage and delivery, unlike a spanning set of vectors or a group of atoms on a lattice.[5]

Unit

A price difference, cash price minus the futures price, in currency per unit of the good

Recovery means

Not applicable

Measured by

Computed as the quoted cash price minus the quoted futures price and tracked over the life of the contract