It is assumed that the student has knowledge of one semester of second year undergraduate linear algebra. After an introduction to finite-dimensional coordinate space, matrices, row equivalence of matrices, determinant of a square matrix, and rank of a matrix with applications to the solution of systems of linear equations, the pre-requisite includes the concepts (over the real field) of vector space, basis, dimension, and linear transformation with almost total emphasis on the finite-dimensional case. In that case linear maps are studied through their matrices. The (trivial) classification of linear maps (by dimension of domain, target, and image) is a consequence of the theory of matrices.
The important notion of similarity (classification of linear endomorphisms) is introduced, the notion of determinant and the characteristic polynomial are found to be similarity invariants. Attempts at the second undergraduate year to attack the problem of similarity do not usually go beyond an introduction to eigenvectors and eigenvalues, the notion of diagonalizability, and the result that a square matrix of size with distinct eigenvalues is diagonalized by a basis of eigenvectors.
Sometimes the pre-requisite course will discuss inner products, orthogonality, the characterization of orthogonal transformations as the distance preserving linear transformations, and perhaps the orthogonal diagonalizability of real symmetric matrices.
MATH 424/524 needs to begin with a quick (perhaps 4 hour) review of the earlier material.
The material is presented in the context of a fixed field of scalars that is not necessarily the real field. This is, on the one hand, essential in order to be able to give a sane treatment of eigenvalues, even the orthogonal diagonalization of real symmetric operators, which requires looking at the complexification of the operator, and, on the other hand, requires no more effort than working over the real field. Moreover, the “rational canonical form” furnishes the solution of the problem of similarity for not only the rational field but, in fact, for every field including the real field and the complex field (where its theory quickly yields that of the Jordan normal form). Slipping into the context of an arbitrary field of scalars should be done as part of the aforementioned review.
Most of the course results are straightforward to prove and might be regarded as exercises for the adventurous instructor though not for the students.
There are five goals:
Provide the student with an understanding that linear algebra and linear geometry are the same subject and with a sensitivity to whether or not questions involve coordinate-free concepts. If it is easier to prove a coordinate-free result by choosing coordinates than by using coordinate-free methods, feel free to do so. This can be a useful research technique.
Introduce the most basic constructions in the category of vector spaces over a field including Hom, direct product, direct sum and, most important, the quotient of a vector space by a linear subspace. Point out that the fact that is a consequence of the fact that the dimension of a space less the dimension of a subspace is the dimension of the corresponding quotient. The quotient is characterized by its universal/initial mapping property. The direct product is characterized by its universal/final mapping property, while the direct sum is universal/initial. (The same statements for the product are true also in the category of sets and the category of topological spaces, while the “direct sum” in these categories is given by “disjoint union”.) For a finite number of factors the direct product and the direct sum of vector spaces are isomorphic.
Provide a full treatment of duality for finite-dimensional vector spaces. The dual of a vector space is the space , where is the field of scalars. It is a contravariant functor of . For any vector space there is a natural linear map from the space to the its second dual, i.e., the dual of its dual. This map is always injective. In the finite-dimensional case it must (by the dimension formula) be an isomorphism. A linear map from one vector space to another admits a natural contravariant map from the dual of the second to the dual of the first. If bases of the given vector spaces are chosen and one chooses then the dual bases of the dual spaces, then the matrix of the contravariant dual is the transpose of the matrix of the original. For finite-dimensional spaces the dual of a short exact sequence is another, and, therefore, the second dual of a short exact sequence is the original. (This type of duality is also encountered in the study of locally compact abelian groups, and is one of the corner stones of Fourier analysis in that context. The only real vector spaces that are locally compact are those that are finite-dimensional, and in that case the two notions of duality coincide.)
Study multi-linear algebra.
The important bilinear case includes the study of quadratic forms and the diagonalization of quadratic forms over a field where . In turn, one finds that a quadratic form in variables over the complex field is determined up to isomorphism (sometimes called equivalence) by and its rank, while over the real field one needs , the rank, and the signature. (The rational case is much more complicated.) If one restricts to orthogonal isomorphism over the real field with respect to a given inner product, then the eigenvalues are invariants. This latter type of classification is equivalent to the orthogonal classification of symmetric linear endomorphisms of the same vector space. The entire theory of the multivariate normal distribution in probability rests on the orthogonal classification of finite-dimensional quadratic forms and on the one-dimensional “probability integral”.
The tensor product of two vector spaces should be defined to be an initial object in the category of bilinear maps emanating from the product of the two spaces. A standard argument shows that it is unique up to unique isomorphism. It may be constructed as the quotient (modulo the subspace of “relations”) of the direct sum (not product) of as many copies of the scalar field as there are elements of the product space. The tensor product of two finite dimensional vector spaces is naturally isomorphic to the dual of the space of bilinear maps from the product to the scalar field.
The tensor product of two vector spaces is “associative” modulo isomorphism, so one may speak of the tensor product of a finite number of vector spaces.
The symmetric powers and the exterior powers are appropriate quotients of the tensor product of finitely many copies of a given vector space. For this course the exterior powers are more important. The determinant of a finite-dimensional linear endomorphism is characterized as the trace of its highest exterior power; this is equivalent to the statement that the determinant of an matrix is the unique alternating n-linear form on row space that assigns the value to the identity matrix. More generally, the coefficients of its characteristic polynomial are the traces of its exterior powers in general.
Treat the classification of linear endomorphisms of a finite-dimensional vector space over an arbitrary field. The result is the following: Let be an matrix over the field . Form the “characteristic matrix” , a matrix of polynomials with coefficients in the field . Then perform row and column operations on this polynomial matrix to bring it into diagonal form with diagonal elements that are monic polynomials which successively divide each other. (There is an algorithm to do this since long division is available in the ring of polynomials.) Then the given matrix is similar to the direct sum of the companion matrices of the polynomials . Moreover, no two of these direct sums of companion matrices are similar. (Observe that the product of the invariant factors is the characteristic polynomial of and their least common multiple is the minimal polynomial of .)
First show that an endomorphism of an -dimensional vector space makes that vector space a module over the ring of polynomials via the formula and that the isomorphism class of (as an endomorphism), which is obviously the same thing as the similarity class of any matrix of , is the same thing as the -linear isomorphism class of as a module over the polynomial ring .
That established, one needs merely to show that the cokernel of the -linear map given by the characteristic matrix is isomorphic as an -module to . This is a modestly challenging exercise for the instructor.
This yields up the Jordan normal form once one considers the decomposition of the successively divisible invariant polynomials into irreducible factors and realizes that the direct sum of the companion matrices of two coprime polynomials is similar to the companion matrix of their product. (This is the “Chinese Remainder Theorem” for polynomials.)
Probably the best way to establish the uniqueness of successively divisible monic invariant factors is to look at the exterior powers of the characteristic matrix. The exterior power construction works as one expects for free modules of finite rank over the polynomial ring and their linear endomorphisms.
The theory of the rational canonical form is a special case of the classification of finitely-generated modules over a principal ideal domain, which is part of Math 520. The treatment here is algorithmic since the polynomial ring admits algorithmic long division. Computer algebra systems often have functions to carry out this algorithm. An algorithmic treatment for general principal ideal domains is not available. Note also that the decomposition of a polynomial into irreducible factors is not algorithmic. That is, the Jordan normal form is not algorithmic.