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    Vector  Calculus

Josiah Willard Gibbs                       

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     Contravariance and Covariance


 Part 2 -- 9/22/2009
           
Vector Concept 2 -- Vector Spaces:
     The axiomatic approach used in algebra requires the introduction of a number of definitions.  However, only the definitions needed to illustrate contravariance and covariance in an abstract vector space will be given:

• Vector Space
• Linearly Independent
• Span
• Basis
• Coordinates
• Change of Basis
• Linear Functional
• Dual Space

     The definition of a vector space also uses the algebraic concept of a field. (Not to be confused with a physical field, like, for example, the gravitational field...same words, different concepts.) For those who have not studied fields and do not wish to, good examples are the real numbers R, or the complex numbers C, which can be substituted in the definition below to allow you to follow along. Boldface is used to indicate elements of the vector space, normal font is used to indicate scalars.

I now introduce some notation and a convention that will be useful.
The Kronecker delta is defined by:
 
Also, the Einstein summation convention will be used. That is, whenever repeated indices appear in an equation, it is understood that a summation is taken over these repeated indices. 


Defninition: A vector space V consists of a set of objects, called vectors, and a field F, called scalars , and a binary operation +  on vectors, that satisfies the following axioms:

i) if a and b are vectors, then so is a +b for all vectors a and b

ii) a+ b =b + a

iii) a + (b + c) = (a + b) + c

iv) There is a zero vector, 0, such that 0 + a = a + 0 = a for any vector a

v) There are additive inverses. For each vector a, there exists a vector -a such that a + (-a) = -a + a = 0.

vi) Scalar multiplication. For any scalar k and any vector a, ka is a vector.
vii) k(a + b) = ka + kb

viii) (k + l)a = ka + la for any scalars k and l

ix) k(l(a)) = kl(a) for any scalars k and l

x) 1a = a1 = a where 1 is the scalar identity


Definition: A set of vectors {v₁,...,v_k} is linearly independent iff
a₁v₁ + ... + a_kv_k = 0 implies a₁=...=a_k= 0

Definition: The span of a set of vectors {v₁,...,v_k} is the intersection of all subspaces that contain {v₁,...,v_k} 

     A straightforward result is that the span of a set of vectors {v₁,...,v_k} is given by all possible linear combinations of the {v₁,...,v_k}. This result is often more useful to work with than the actual definition.

Definition: A basis for a vector space is any set of linearly independent vectors that span V.

     Given a basis for V, the number of vectors in the basis is called the dimension of V. A finite dimensional vector space is a vector space that is spanned by a finite number of vectors. The discussion will be limited to finite dimensional vector spaces.

Definition: Let v be a vector in V, and let β = {v₁,...,v_n} be an ordered basis for V. Then if v = a¹v₁ + ... + aⁿv_n , the scalars a¹,...,aⁿ are called the coordinates of v with respect to the basis β, and is written as a nx1 column matrix:

Change of Basis

We do not wish to be constrained to one basis. If we formulate an equation that uses coordinates in its formulation, we would like to know that it is true independent of the basis chosen. Therefore, we need to know how to express our equation in a new basis. Also, it is often the case that we are able to transform to a new basis where our equations take on a simpler form. For these reasons, it important to study the change of basis problem.

Let


 
be a basis for V and v be an arbitrary vector with coordinates 

If we now instead wish to use a different basis



how do we calculate the new coordinates?

To answer  this question, expand v in the beta basis, using the ccordinates just given:

Then, writing this in coordinate form in the alpha basis gives

This can be re-written as a nxn matrix times a nx1 matrix as



where the nxn matrix's columns are the coordinates of the beta basis vectors with respect to the alpha basis. Denoting the nxn matrix, by P, this can be written as
                         v_α = P v_β

and left multipling by P^-1 gives the require expression for the coordinates of v in the new basis:

                         v_β = P^-1 v_α

The matrix P is called the transistion matrix from the α basis to the β basis.
( A word on convention; if instead of expanding v in the β basis above, I had expanded v in the α basis, and taken coordinates in the β basis, I would have finally arrived at a slightly different form:
                          v_β = Q v_α
where Q is the matrix formed by the coordinate vectors of the α basis vectors with respect to the β basis. Occasionally, people call Q the transition matrix. If you are reading a book on linear algebra you need to check the convention being used.)


Recall in the "directed arrow" definition of a vector,  the contravariant components of a vector were its coordinates with respect to the reciprocal lattice, and the reciprocal lattice was defined by the requirement e^i.e_k = 0. In the abstract vector space formulation, this notion is generalized by the concept of the dual space and dual basis, which we now introduce.

Definition: A linear functional on a vector space V is a linear map from V into the scalars K.

Definition: The set of all linear functionals on a vector space V is called the dual space of V, denoted by V^*. Elements of the dual space are called dual vectors.

If V is a vector space of dimension n, then V^* is a vector space of dimension n. Given a basis α for V, we construct a basis for V^* as follows:

Definition: For a given basis α ,



the set of linear functionals

 
defined by

 is a basis for V*, called the dual basis of α.


We now know how to get a basis for V^* given a basis for V . This allows us to write the components for an aritrary dual vector. We now want to enquire how the coordinates of our arbitary dual vector change when a change of basis is performed is performed in V. The change of basis induces a change of basis on the dual space.
              

Old Bases	New Bases

The key to discovering how dual vectors transform is to require

Let the transformation matrix to the new dual basis be h_ik .

Then gⁱ=h_ik f^k, where the Einstein summation convention is used

to indicate a summation is taken over the indices k. If β_j = Q_jlα_l

then we have  h_ik f^kQ_jlα_l  = δⁱ_j . Using the summation convention

and simplifying this reduces to h_ik Q_jl  δ^k_l = δⁱ_j  and simplifying one more

time gives  h_il Q_jl= δⁱ_j 

or in other words h_il Q^T_lj= δⁱ_j 

where Q^T is the transpose of Q.
Thus we have the important result that h= (Q^T)^-1
So, the coordinates in the new dual basis are computed from the change of basis matrix of the by taking the transpose inverse of the transformation matrix and using that as the transformation matrix for the dual vectors. The dual vectors are also called contravariant vectors, and the vectors in the original space are called covariant vectors. The contravariant vectors are said to transform contragrediently and the covariant vectors transform cogrediently.
 
Next week -- Tensor analysis and vector transformation laws.

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