Appendix C: Vectors, Matrices, Orthogonal Functions
Paul G. Huray · 2010
Vectors, Matrices, Orthogonal FunctionsWhen using higher dimensions, it is easier to use a numerical designation for the Cartesian coordinate axes, as shown in Figure С.1.In Figure C.l, the names of the base vectors â" â y , â : have been replaced with â\, â 2 , â 3 , and the names of the components of the vector Л along the x u x 2 , x$ coordinate axes have been replaced with А ь A 2 , Α λ .This convention is convenient for writing the vector À aswhere the summation convention has been used such that when an index subscript, i, is repeated, the sum over i is implied.This is shorthand in three dimensions, but it is crucial when we want to express a vector A in four dimensions or higher where it is impossible to visualize the four components of A but for which the algebra continues to apply. /i-DIMENSIONAL (w-D) VECTOR SPACEIn an n-D vector space, there are n linearly independent vectors in the space but not n + I. Linear independence means that, for the n vectors in the set x-" there is no set II of coefficients c, such that ^c,x, =0 except the set c, = 0.If a set of vectors a, forms a basis' (coordinate system) for a vector space, there exists a set of numbers II х-, such that x = V χ,α, for any arbitrary vector л: in the space.1 A basis set is any set of vectors a. that "spans the space"; that is.given an arbitrary vector in the space, we may find a set of Xj so that x = } χ,ύ,.