Mathematical Methods of Physics/Vector Spaces
As is no doubt seen in elementary Physics, the notion of vectors, quantities that have a "magnitude" and a "direction" (whatever these may be) is very convenient in several parts of Physics. Here, we wish to put this idea on the rigorous foundation of Linear Algebra, to facilitate its further use in Physics. The interested reader is encouraged to look up the Wikibook Linear Algebra for details regarding the intricacies of the topic.
Let be field and let be a set. is said to be a Vector Space over along with the binary operations of addition and scalar product iff
(iii) such that ...(Identity)
(iv) such that ...(Inverse)
The elements of are called vectors while the elements of are called scalars. In most problems of Physics, the field of scalars is either the set of real numbers or the set of complex numbers .
Examples of vector spaces:
(i) The set over can be visulaised as the space of ordinary vectors "arrows" of elementary Physics.
(ii) The set of all real polynomials is a vector space over
(iii) Indeed, the set of all functions is also a vector spaces over , with addition and scalar multiplication defined as is usual.
Although the idea of vectors as "arrows" works well in most examples of vector spaces and is useful in solving problems, the latter two examples were deliberately provided as cases where this intuition fails to work.
A set is said to be linearly independant if and only if
implies that , whenever
A set is said to cover if for every there exist such that . (we leave the question of finiteness of the number of terms open at this point)
A set is said to be a basis for if is linearly independant and if covers .
If a vector space has a finite basis with elements, the vetor space is said to be n-dimensional
As an example, we can consider the vector space over reals. The vectors form one of the several possible basis for . These vectors are often denoted as or as
Let be a vector space and let be a basis for . Then any subset of with elements is linearly independant.
By definition of basis, there exist scalars such that
Hence we can write as that is
Which has a nontrivial solution for . Hence is linearly dependant.
If a vector space has a finite basis of elements, we say that the vector space is n-dimensional
An in-depth treatment of inner-product spaces will be provided in the chapter on Hilbert Spaces. Here we wish to provide an introduction to the inner product using a basis.
Let be a vector space over and let be a basis for . Thus for every member of , we can write . are called the components of with respect to the basis .
We define the inner product as a binary operation as , where are the components of with respect to
Note here that the inner product so defined is intrinsically dependant on the basis. Unless otherwise mentioned, we will assume the basis while dealing with inner product of ordinary "vectors".
Let , be vector spaces over . A function is said to be a Linear transformation if for all and if
Now let and be bases for respectively.
Let . As is a basis, we can write .
Thus, by linearity we can say that if , we can write the components of in terms of those of as
The collection of coefficients is called a matrix, written as
and we can say that can be represented as a matrix with respect to the bases
Let be a vector space over reals and let be a linear transformations.
Equations of the type , to be solved for and are called eigenvalue problems. The solutions are called eigenvalues of while the corresponding are called eigenvectors or eigenfunctions. (Here we take )