Linear space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Vector_space


00:03:49 1 Introduction and definition
00:04:07 1.1 First example: arrows in the plane
00:05:30 1.2 Second example: ordered pairs of numbers
00:06:06 1.3 Definition
00:06:55 1.4 Alternative formulations and elementary consequences
00:07:08 2 History
00:09:11 3 Examples
00:11:17 3.1 Coordinate spaces
00:12:46 3.2 Complex numbers and other field extensions
00:16:25 3.3 Function spaces
00:16:35 3.4 Linear equations
00:17:20 4 Basis and dimension
00:17:32 5 Linear maps and matrices
00:17:38 5.1 Matrices
00:17:50 5.2 Eigenvalues and eigenvectors
00:19:50 6 Basic constructions
00:21:13 6.1 Subspaces and quotient spaces
00:21:31 6.2 Direct product and direct sum
00:21:56 6.3 Tensor product
00:22:30 7 Vector spaces with additional structure
00:23:11 7.1 Normed vector spaces and inner product spaces
00:23:33 7.2 Topological vector spaces
00:24:41 7.2.1 Banach spaces
00:25:24 7.2.2 Hilbert spaces
00:28:58 7.3 Algebras over fields
00:29:38 8 Applications
00:30:08 8.1 Distributions
00:32:30 8.2 Fourier analysis
00:33:38 8.3 Differential geometry
00:36:36 9 Generalizations
00:37:29 9.1 Vector bundles
00:38:45 9.2 Modules
00:39:21 9.3 Affine and projective spaces
00:40:58 10 See also
00:45:34 11 Notes
00:45:58 12 Citations
00:46:05 13 References
00:47:58 13.1 Algebra
00:49:11 13.2 Analysis
00:50:06 13.3 Historical references
00:50:26 13.4 Further references
00:51:52 14 External links
00:52:03 Normed vector spaces and inner product spaces
00:57:33 Topological vector spaces
01:02:30 Banach spaces
01:12:00 Hilbert spaces
01:15:52 Algebras over fields
01:17:31 0 (Jacobi identity).Examples include the vector space of n-by-n matrices, with [x, y]
01:18:50 − v2 ⊗ v1 yields the exterior algebra.When a field, F is explicitly stated, a common term used is F-algebra.
01:19:03 Applications
01:19:13 Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods. Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.
01:20:08 Distributions
01:21:20 {p}, the set consisting of a single point, this reduces to the Dirac distribution, denoted by δ, which associates to a test function f its value at the p: δ(f)
01:22:43 Fourier analysis
01:27:55 Differential geometry
01:29:19 Generalizations
01:29:29 Vector bundles
01:32:28 Modules
01:33:43 Affine and projective spaces
01:34:48 bgeneralizing the homogeneous case b
01:35:43 See also
01:35:58 Notes



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SUMMARY
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A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.
Euclidean vectors are an example of a vector space.  They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dim ...