|
In mathematics, an affine combination of vectors x1, ..., xn is a linear combination Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
 in which the sum of the coefficients is 1, thus:  . Here the vectors are supposed to lie in given vector space V over a field K; and the coefficients αi are scalars in K. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...
This concept is important, for example, in Euclidean geometry. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...
An affine combination of fixed points of an affine transformation is also a fixed point, so the fixed points form an affine subspace (in 3D: a line or a plane, and the trivial cases, a point and the whole space). In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...
Motivation
This article or section is not written in the formal tone expected of an encyclopedia article.
Please improve it or discuss changes on the talk page. See Wikipedia's guide to writing better articles for suggestions. Suppose points in space are associated with vectors. Imagine that Smith knows that a certain point is the origin, and Jones believes that another point -- call it p -- is the origin. Two points, a and b are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find a point that he thinks is a + b, but is actually p + (a − p) + (b − p) = p + (a + b). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However -- and note this well: In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
- If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!
The proof is a routine exercise. Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure" -- i.e., the values of affine combinations.  There is no number other than 1 with which the same idea works.
See also |
Feeds |
Contact