Real vector space | Wikipedia audio article |
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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Vector_space 00:03:45 1 Introduction and definition 00:04:03 1.1 First example: arrows in the plane 00:05:28 1.2 Second example: ordered pairs of numbers 00:06:04 1.3 Definition 00:06:52 1.4 Alternative formulations and elementary consequences 00:07:05 2 History 00:09:11 3 Examples 00:11:19 3.1 Coordinate spaces 00:12:49 3.2 Complex numbers and other field extensions 00:16:26 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 00:17:49 5.2 Eigenvalues and eigenvectors 00:19:48 6 Basic constructions 00:21:10 6.1 Subspaces and quotient spaces 00:21:27 6.2 Direct product and direct sum 00:21:52 6.3 Tensor product 00:22:26 7 Vector spaces with additional structure 00:23:06 7.1 Normed vector spaces and inner product spaces 00:23:27 7.2 Topological vector spaces 00:24:35 7.2.1 Banach spaces 00:25:18 7.2.2 Hilbert spaces 00:28:52 7.3 Algebras over fields 00:29:31 8 Applications 00:30:00 8.1 Distributions 00:32:21 8.2 Fourier analysis 00:33:29 8.3 Differential geometry 00:36:25 9 Generalizations 00:37:18 9.1 Vector bundles 00:38:34 9.2 Modules 00:39:10 9.3 Affine and projective spaces 00:40:47 10 See also 00:45:22 11 Notes 00:45:45 12 Citations 00:45:52 13 References 00:47:43 13.1 Algebra 00:48:57 13.2 Analysis 00:49:53 13.3 Historical references 00:50:13 13.4 Further references 00:51:41 14 External links 00:51:52 Normed vector spaces and inner product spaces 00:57:14 Topological vector spaces 01:02:13 Banach spaces 01:11:32 Hilbert spaces 01:15:22 Algebras over fields 01:16:56 0 (Jacobi identity).Examples include the vector space of n-by-n matrices, with [x, y] 01:18:15 − v2 ⊗ v1 yields the exterior algebra.When a field, F is explicitly stated, a common term used is F-algebra. 01:18:28 Applications 01:18:37 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. 01:19:33 Distributions 01:20:44 {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) 01:22:08 Fourier analysis 01:27:21 Differential geometry 01:28:46 Generalizations 01:28:56 Vector bundles 01:31:52 Modules 01:33:05 Affine and projective spaces 01:34:09 bgeneralizing the homogeneous case b 01:35:03 See also 01:35:19 Notes Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts Speaking Rate: 0.7370033346060955 Voice name: en-AU-Wavenet-C "I cannot teach anybody anything, I can only make them think." - Socrates SUMMARY ======= 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-dime ... |