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Linear algebra

Linear algebra. What is it all about? What is its use? Starting with such concepts as a "linear space", "inner product", "linear transformation", etc. defined from sets of axioms, one develops a body of theorems and results from those axioms that is applicable to anything that meets the basic abstract requirements.

Linear algebra employs the theory and concepts of Matrix Theory but applies them to a much broader assortment of objects than the matrices and vectors encountered in Matrix Theory -- applying them to things like differential and integral equations, infinite series, polynomials, continuous functions, etc.. The concepts, structures and results of Matrix Theory serve as a kind of formal model for abstract structures that are possessed by a broad and diverse assortment of other things. Linear Algebra is a generalization of Matrix Theory that includes "vectors" of a completely different nature from the usual concept of the vector encountered in Matrix Theory. The vector space of Matrix Theory consists of the familiar vectors of one, two, three and n-dimensional space. The "vectors" of Linear Algebra correspond to such widely different things as continuous functions, infinite series, infinite sequences, polynomials, etc.. These things also have their spaces, subspaces, bases, etc.

In Linear Algebra, starting from abstract, axiomatic definitions for a linear space and a linear transformation one proceeds to such non-trivial ideas as eigenvectors and eigenvalues associated with linear transformations -- ideas important not only in Matrix Theory but also in other areas (such as the solution of differential and integral equations).