here is a primer of linear algebra:
the idea is to study phenomena where the output varies in the same way as the input. e.g. derivatives. the derivative of a sum is the sum of the derivatives, and the derivative of a scalar multiple is that scalar multiple of the derivative.
thus if you know that sin' = cos and (cos)' = -sin, then you also know that
(asin+bcos)' = acos(x) - bsin. this gives you an infinite 2 dimensional family of solutions, generated by just two "independent" solutions to the same problem.
as a result you can systematize this process of taking derivatives for this family of functions into a "matrix" of numbers. i.e. we can represent asin+bcos, just by (a,b), and the fact that asin +bcos goes to -bsin + acos, by the 2x2 matrix with rows
[0 -1] and [1 0]. i.e. to get the two coefficients (-b,a) of the output -bsin + acos, you just dot the coefficient vector (a,b) of the input with the two row vectors [0 -1] and [1 0]. Many other operations have this same matrix. e.g. rotation of the plane aboiut the origin also takes vector sums to vector sums, and scalar multiples to scalar multiples. This matrix above also is the matrix for a 90 degree rotation clockwise.
so linear algebra and matrix theory throw out what the operations are and remembers only their properties, i.e. simply that L is lienar if L(x+y) = L(x)+ L(y) and L(cx) = cL(x).
Then we ask how much we can say about such operations. can we classify them?
all two dimensional spaces of inputs can be regarded as essentiaslly the same as the plane, and so all linear operations on all two dimensional spaces can be regarded as essentially operations on vectors in the plane. if there is any way to assign a notion of "length" to those objects in a 2 diemnsional space such that the given linear operation preserves that length, then in fact the linear operation is essentially equivalent either to a rotation, or to a reflection, or some combination of these. this is essentially the same as saying the matrix for the operation has its inverse equal to its transpose.
so there are two ideas in linear algebra, one is dimension, and the study of whether an operation preserves dimension or diminishes it, and the consequent theory of adding a general solution to the homogeneous problem Lx = 0, to any particular solution of the special problem Lx=b to get a general solution to the specific problem, the second is the study of the structure of operations that preserve dimension, i.e. isomorphisms.
since the simplest linear operations are scalar multiplication, the first task of classification is to understand all those linear op[erations that are composed of sums of these simplest operations, the operations whose matrices are diagonalizable. in this situation each basis vector is acted on by the operation via scalar multiplication, and such basis vector are called eigenvectors. that's all they are, objects on which the operation acts via scalar multiplication. If there are enough of them to express all other vectors, then the operation is diagonalizable.
the usual condition guaranteeing this is some kind of notion of angle and then the assumption that the operation is angle preserving. over the reals this does not quite do it and you have to allow also rotations, but over the complexes rotations also are scalar multiples.
so there are two basic theorems in linear algebra. the most basic is the dimension theorem: if V is a space of dimension n, and L:V-->V is a linear operator, then the dimension of the objects y such that Lx=y has a solution, equals n minus the dimension of the space of those x such that Lx=0. then for any such y, if Lu=y say. and Lx=0, then Lx+u = y also, and all solutions of Lw=y have form u+somex with Lx=0.
the second basic theorem is that if the matrix of an operation on some n dimensional space is symmetric about the main diagonal, then there is a basis of eigenvectors.
in the tables of contents above, for mirsky and shilov, the sections called "canonical forms" are about finding the best way to decompose an operation into simpler ones, and the sections on "quadratic" or "hermitian" forms, are about real and complex notions of length and angles, so you can state and prove the results telling when that operation is diagonalizable.
the determinant is a very useful device for determining the eigenvectors of L, in a finite dimensional space. it yields a polynomial called the characteristic polynomial whose roots are those scalars that occur as the scalar multiplier for some eigenvector. i.e. there is a vector v such that Lv = cv if and only if c is a root of the characteristic polynomial.
so there are three levels of linear algebra. 1st they tell you about dimension and rank of matrices. second they define eigenvalues and tell you about operations whose matrices can be chosen to be diagonal. finally the third level tells you what the matrices can be chosen to look like even when they cannot be made diagonal. there are two answers to this last question.
1) over any field there exists a basis made up of blocks like v1,...vn, such that the operator acts "cyclically" on each block: i.e. such that Lv1 = v2, Lv2 = v3,...Lvn-1 = vn, but then it breaks down and we can only say that
Lvn = some linear combination of the v1,...vn.
the associated "companion matrix" has all the first n-1 columns looking just like all zeroes except for one 1 below the main diagonal, and the last column is the coefficient vector of that arbitrary linear combination.
this is called "rational canonical form".
2) over an algebraically closed field, there is a basis made up of blocks of form v1,...,vn such that these are almost eigenvectors. i.e. for each such block there is a number c such that we do not quite have Lvi = cvi, but we do have Lv1 = cv1, then Lv2 = v1 + cv2, Lv3 = v2 + cv3, ...,Lvn = vn-1 + cvn.
then the corresponding matrix block has all c's on the diagonal, but also has 1's just above the main diagonal.
this form is called jordan canonical form.
that pretty much covers the entire linear algebra curriculum. the first hurdle is to understand the concept of dimension, or equivalently to understand linear independence.
the next chapter is multilinear algebra. i.e. quadratic and hermitian forms are actually bilinear forms, in that you multiply two vector together and get a number, in a somewhat symmetric way. the determinant is an n linear alternating form, in that you multiply n vectors in an n dimensional space together and get a number, in an antisymmetric way. then the general concept of multiplying together any finite number of vectors and getting a vector or number, without assuming symmetry or antisymmetry, is called multilinear algebra, or tensor algebra. there is thread devoted to families of these tensors, which come up in differential geometry and hence modern (i.e. post einstein) physics.
then in algebraic geometry and complex manifolds, these families of tensors, or tensor fields, are themselves considered as objects in a sheaf, and there are cohomology groups defined with coefficients in these sheaves, and this tool is called global linear algebra. cohomology groups give a way of mapping certain problems linearly into a space of obstructions, such that the solvable problems are the ones that map to zero. then dimension theory can sometimes give easy criteria for solvability of the problem, for instance if the space of obstructions has lower dimension than the space of problems. for instance on a compact riemann surface of genus g the space of obstructions to the mittag leffler problem has dimension g, so any system of polar data of degree greater than g admits a meromorphic function having poles in that system.
