Foundations of linear algebra

In linear algebra one studies three kinds of objects; matrices, linear spaces,

and algebraic forms. The theories of these objects are so closely related

that most problems of linear algebra have equivalent formulations in each

of the three theories. The matrix point of view, which underlies the present

exposition is the one best adapted to actual calculations. On the other hand,

most problems of linear algebra that arise in geometry and mechanics lead

to algebraic forms, while the best understanding of the internal connections

between different problems of linear algebra is obtained by means of linear

spaces. Therefore the ability to pass from are type of formulation to another

is one of the most important skills to acquire in the study of linear algebra.

From the point of view of the theory of forms, linear algebra falls

naturally into three parts: the theories of linear forms, of bilinear and

quadratic forms, and of multilinear forms. Linear algebra proper usually

encompasses linear and bilinear forms, and the very beginnings of the theory

of multilinear forms as tensor algebras. The more delicate questions of the

theory of multilinear forms belong to the theory of invariants and are not

included in this book.

Linear algebra is a branch of mathematics as old as mathematics itself.

The solving of the equation ax + b = 0 may be considered the original problem

of this subject. Although this problem presents no difficulty, the method

which solves it, together with the properties of the corresponding linear

function = ax + b, are the initial models for the ideas and methods of all of linear algebra. For example, the fundamental idea behind the solution of

a system of linear equations in several unknowns is that of replacing such

a system by a chain of these simple equations.

The study of systems of linear equations acquired new significance after the

creation of analytic geometry; it was possible to reduce all the fundamental

questions about the