Basis - The list of  properties or entities (such as vectors) which are used to practically describe an object or system or its state.

Object Example - For example, an object which has a mass of 10 kg, at a temperature of 35 degrees Celsius, made of steel and is blue in color may be described with the state descriptor (10, 35, AISI 1020 steel, blue) if the state basis has been defined as (mass in kg, temperature in degrees Celsius, material identifier, surface color).  The identifier 'AISI 1020 steel' also implies other properties such as density and hardness which may or may not be relevant.

Positional Coordinate System - The basis of a positional rectilinear reference coordinate system (RCS) uses spatial vectors  (usually referred to as i,j, and k) which can have any dimensions such as millimeters or inches. This RCS requires a stated origin point from which the direction and distance from the origin for an object can be determined by a linear combination of i,j, and k ( ai+b,j+ck). See tutorial - Reference Coordinate Systems and Transformations.