LINEAR COMBINATIONS

A Simple Analogy

Linear combinations of basis vectors are used to represent arbitrary vectors. This concept appears in many areas of science and mathematics and is important in parametric curve representation.

A simple example is the concept that any point in "space" (which can also be viewed as a vector from the origin of an arbitrary coordinate system to the point) can be represented by a linear combination of mutually orthogonal unit vectors – one in each of the x, y, and z directions. The direction of the unit vectors as well as the location of the origin are arbitrary, but, of course, change the coefficients of the vectors. These coefficients are often named coordinates. It is important that the unit vectors are linearly independent: none can be written as a linear combination of the others. It is not possible to create a vector in the z direction by any combination of the other vectors in the basis – one in the x direction and one in the y direction.

A simple analogy is a recipe for a cake:

If the constants a, b, . . . , i are allowed to represent the required amounts of the ingredients, the recipe can be stated as:

cake = a * flour + b * sugar + c * cocoa + d * soda + e * salt + f * water + g * oil + h * vinegar + i * vanilla

Note that the ingredients are linearly independent in that it is impossible to make water by any linear combination of the other ingredients. However, the ingredients do not form a basis for cakes in the sense that some cakes would require other ingredients. If other ingredients were part of our basis, they could be represented with 0 coefficients. Assuming that the only other possible ingredient for a cake is eggs, the basis could be stated as (in alphabetical order)

(cocoa, eggs, flour, oil, salt, soda, sugar, vanilla, vinegar, water)

and the above cake could be referenced as (ignoring measurement labels)

( .25, 0, 1.5, .25, .5, 1, .75, 1, 1, 1 ).