linear algebra (original) (raw)

Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering. Diverse disciplines, such as differential equations,differential geometry, the theory of relativity, quantum mechanics, electrical circuits, computer graphics, and information theory benefit from the notions and techniques of linear algebra.

Euclidean geometry is related to a specialized branch of linear algebra that deals with linear measurement. Here the relevant notions are length and angle. A typical question is the determination of lines perpendicular to a given plane. A somewhat less specialized branch deals with affine structure, where the key notion is that of area and volume. Here determinants play an essential role.

The following subject outline surveys key topics in linear algebra.

1. 1. Linear structure.
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      Linear mappings: linearity axioms, kernels and images, injectivity, surjectivity, bijections, compositions, inverses,matrix representations, change of bases, conjugation, similarity.
2. 1. Affine structure.
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      Geometric aspects: Euclidean volume, orientation, equiaffine transformations, determinants as geometric invariantsof linear transformations.
3. 1. Diagonalization and Decomposition.
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6. 1. Computational and numerical methods.
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      Eigenvalue problems: singular value decomposition, Gauss and Jacobi-Siedel iterative algorithms.