Notes on Linear Algebra

A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field). In each field F there exist unique elements called o and 1 which, under the operations of addition and multiplication, behave with respect to all the other elements of F exactly as their correspondents in the real number system. In two respects, the analogy is not complete: 1) multiplication is not assumed to be commutative in every field, and 2) a field may have only a finite number of elements.

A binary operation φ on a set A is a function from A × A to A, i.e., φ : A × A → A. φ((x, y)) is also denoted by xφy. Often we use a symbol for φ: +, ·, − , .

Springer Undergraduate Mathematics Series, 2002

Peano also postulated the existence of a zero object 0 and used the notation a-b for a + (-b). By introducing the notions of dependent and independent objects, he defined the notion of dimension, showed that finite-dimensional spaces have a basis and gave examples of infinite-dimensional linear spaces. If one considers only functions of degree n, then these functions form a linear system with n + 1 dimensions, the entire functions of arbitrary degree form a linear system with infinitely many dimensions. Peano also introduced linear operators on a linear space and showed that by using coordinates one obtains a matrix. With the passage of time, much concrete has set on these foundations. Techniques and notation have become more refined and the range of applications greatly enlarged. Nowadays Linear Algebra, comprising matrices and vector spaces, plays a major role in the mathematical curriculum. Notwithstanding the fact that many important and powerful computer packages exist to solve problems in linear algebra, it is our contention that a sound knowledge of the basic concepts and techniques is essential.

Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the most fundamental topics.