Notes on Linear Algebra∗
نویسنده
چکیده
Linear algebra provides the foundational setting for the study of multivariable mathematics which in turn is the bedrock upon which most modern theories of mathematical physics rest including classical mechanics (rigid body mechanics), continuum mechanics (the mechanics of deformable material bodies), relativistic mechanics, quantum mechanics, etc. At the heart of linear algebra is the notion of a (linear) vector space which is an abstract mathematical structure introduced to make rigorous the classical, intuitive concept of vectors as physical quantities possessing the two attributes of length and direction. In these brief notes, vector spaces and linear transformations are introduced in a three step presentation. First they are studied as algebraic objects and a few important consequences of the concept of linearity are explored. Next the algebraic structure is augmented by introducing a topological structure (via a metric or a norm, for example) providing a convenient framework for extending key concepts from the calculus to the vector space setting permitting a rigorous framework for studying nonlinear, multivariable functions between vector spaces. Finally, an inner product structure for vector spaces is introduced in order to define the geometric notion of angle (and as a special case, the key concept of orthogonality) augmenting the notion of length or distance provided by previously by a norm or a metric.
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