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-in O
-fluid B-Attributes
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-is O
-integral O
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-fluid O
-
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-
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-
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-or O
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-emphasize O
-that O
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-a O
-function O
-space O
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-is O
-a O
-common O
-convention O
-in O
-functional O
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-to O
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-case O
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-have O
-different O
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-for O
-different O
-conventions O
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-be O
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-emphasize O
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-and O
-are O
-real O
-vector O
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-needed O
-or O
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-can O
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-emphasize O
-that O
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-a O
-function O
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-a O
-common O
-convention O
-in O
-functional B-Math
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-a O
-common O
-convention O
-in O
-functional O
-analysi O
-
-a O
-vector B-Math
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-over O
-a O
-field O
-f O
-is O
-a O
-nonempty O
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-a O
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-over O
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-field B-Math
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-a O
-nonempty O
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-a O
-binary O
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-and O
-a O
-binary O
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-eight O
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-nonempty B-Math
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-a O
-binary O
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-a O
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-belo O
-
-linear B-Math
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-nonlinear O
-realworld O
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-makes O
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-functional B-Math
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-
-in O
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-linear B-Math
-mappings I-Math
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-a O
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-
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-makes O
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-triple O
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-vali O
-
-integer B-Math
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-a O
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-of O
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-
-very O
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-in O
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-in O
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-fiel O
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-in O
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-fiel O
-
-if O
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-has O
-a O
-basis B-Math
-of O
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-an O
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-a O
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-size O
-of O
-n O
-elements O
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-an O
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-is O
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-by O
-a O
-square B-Math
-matrix I-Math
-of O
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-
-not O
-all O
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-to O
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-
-bases B-Attributes
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-element O
-of O
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-a O
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-elements O
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-every O
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-of O
-v O
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-be O
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-in O
-a O
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-as O
-a O
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-combination I-Math
-of O
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-
-otherwise O
-a O
-differential B-Math
-equation I-Math
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-is O
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-the O
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-derivative O
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-is O
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-homogeneous B-Math
-function I-Math
-of O
-the O
-unknown O
-function O
-and O
-its O
-derivative O
-
-in O
-this O
-article O
-vectors B-Math
-are O
-represented O
-in O
-boldface O
-to O
-distinguish O
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-from O
-scalar O
-are O
-represented O
-in O
-boldface O
-to O
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-from O
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-
-in O
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-a O
-system O
-with O
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-equations B-Math
-than O
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-infinitely O
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-solutions O
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-may O
-have O
-no O
-solutio O
-
-it O
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-shown O
-that O
-the O
-two O
-approaches O
-are O
-essentially O
-equivalent B-Math
-
-most O
-of O
-the O
-theory O
-of O
-abelian B-Math
-groups I-Math
-may O
-be O
-extended O
-to O
-modules O
-over O
-a O
-principal O
-ideal O
-domai O
-may O
-be O
-extended O
-to O
-modules B-Math
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-a O
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-ideal O
-domai O
-
-nearly O
-all O
-scientific O
-computations O
-involve O
-linear B-Math
-algebra I-Math
-
-the O
-modeling O
-of O
-ambient O
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-is O
-based O
-on O
-geometry B-Math
-
-in O
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-new O
-at O
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-geometry B-Math
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-geometry O
-points O
-are O
-represented O
-by O
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-are O
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-of O
-three O
-real O
-numbers O
-in O
-the O
-case O
-of O
-the O
-usual O
-threedimensional O
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-now O
-called O
-cartesian B-Math
-geometry I-Math
-points O
-are O
-represented O
-by O
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-coordinates O
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-are O
-sequences O
-of O
-three O
-real O
-numbers O
-in O
-the O
-case O
-of O
-the O
-usual O
-threedimensional O
-spac O
-
-for O
-more O
-details O
-see O
-linear B-Math
-equation I-Math
-over O
-a O
-rin O
-over O
-a O
-ring B-Math
-
-hadamard O
-also O
-founded O
-the O
-modern O
-school O
-of O
-linear B-Math
-functional I-Math
-analysis I-Math
-further O
-developed O
-by O
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-and O
-the O
-group O
-of O
-polish O
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-around O
-stefan O
-banac O
-
-to O
-have O
-a O
-vector B-Math
-space I-Math
-the O
-eight O
-following O
-axioms O
-must O
-be O
-satisfied O
-for O
-every O
-u O
-v O
-and O
-w O
-in O
-v O
-and O
-a O
-and O
-b O
-in O
-
-these O
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-the O
-most O
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-spaces I-Math
-with O
-scalars O
-in O
-an O
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-field O
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-are O
-also O
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-field O
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-also O
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-
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-is O
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-of O
-mathematical O
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-the O
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-of O
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-kind O
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-or O
-topology O
-and O
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-respecting O
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-is O
-a O
-branch O
-of O
-mathematical B-Math
-analysis I-Math
-the O
-core O
-of O
-which O
-is O
-formed O
-by O
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-study O
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-kind O
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-example O
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-product O
-norm O
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-topology O
-and O
-the O
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-functions O
-defined O
-on O
-these O
-spaces O
-and O
-suitably O
-respecting O
-these O
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-the O
-core O
-of O
-which O
-is O
-formed O
-by O
-the O
-study O
-of O
-vector B-Math
-spaces I-Math
-endowed O
-with O
-some O
-kind O
-of O
-limitrelated O
-structure O
-for O
-example O
-inner O
-product O
-norm O
-or O
-topology O
-and O
-the O
-linear O
-functions O
-defined O
-on O
-these O
-spaces O
-and O
-suitably O
-respecting O
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-structure O
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-some O
-kind O
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-limitrelated O
-structure O
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-example O
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-and O
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-and O
-suitably O
-respecting O
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-norm B-Math
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-the O
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-on O
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txt_data/train.txt

@@ -1,6963 +0,0 @@
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-a O
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-
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-
-in O
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-
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-
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-quadratic O
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-
-in O
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-involve O
-
-the O
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-trichotomy O
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-case O
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-two O
-variables B-Math
-
-vectors B-Math
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-role O
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-calculus B-Math
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-topology I-Math
-
-a O
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-combination I-Math
-of O
-x O
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-the O
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-expression B-Math
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-
-to O
-express O
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-a O
-vector O
-space O
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-is O
-a O
-linear B-Math
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-of O
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-one O
-commonly O
-uses O
-the O
-following O
-phraseseither O
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-spans O
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-a O
-spanning O
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-vector B-Math
-space I-Math
-v O
-is O
-a O
-linear O
-span O
-of O
-a O
-subset O
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-
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-
-f O
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-linear B-Math
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-is O
-a O
-bijection O
-then O
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-is O
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-then O
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-a O
-linear B-Math
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-then O
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-also O
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-linear O
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-linear B-Math
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-whose O
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-together O
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-multiplied O
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-whose O
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-may O
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-added O
-together O
-and O
-multiplied O
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-the O
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-the O
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-the O
-more O
-general O
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-of O
-modules O
-over O
-a O
-ring O
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-module B-Math
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-
-because O
-a O
-solution O
-to O
-a O
-linear B-Math
-system I-Math
-must O
-satisfy O
-all O
-of O
-the O
-equations O
-the O
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-set O
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-the O
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-
-for O
-improving O
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-them O
-configure O
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-automatically O
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-the O
-theories O
-of O
-measure B-Math
-integration I-Math
-and O
-probability O
-to O
-infinite O
-dimensional O
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-also O
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-infinite O
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-infinite O
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-functional B-Math
-analysis I-Math
-is O
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-the O
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-
-functional B-Math
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-is O
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-quantum O
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-partial B-Math
-differential I-Math
-equations I-Math
-digital O
-signal O
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-electrical O
-engineerin O
-
-for O
-example O
-the O
-navierstokes O
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-techniques O
-derived O
-from O
-linear B-Math
-algebra I-Math
-
-analysis B-Math
-may O
-be O
-distinguished O
-from O
-geometry O
-however O
-it O
-can O
-be O
-applied O
-to O
-any O
-space O
-of O
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-objects O
-that O
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-a O
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-of O
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-may O
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-geometry B-Math
-however O
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-space O
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-
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-linear B-Math
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-
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-linear B-Math
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-
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-
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-linear B-Math
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-matrix B-Attributes
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-
-square B-Math
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-
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-
-vector B-Math
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-
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-are O
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-geometric O
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-of O
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-degree O
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-x O
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-basis B-Attributes
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-linearly B-Math
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-
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-category B-Math
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-linear O
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-are O
-the O
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-vector O
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-linear B-Math
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-are O
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-vector O
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-vector B-Math
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-a O
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-linear B-Math
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-relatively O
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-a O
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-is O
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-
-the O
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-of O
-these O
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-dimension B-Math
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txt_data/valid.txt

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-of O
-the O
-vectors O
-in O
-s O
-for O
-example O
-two O
-linearly O
-independent O
-vectors O
-span O
-a O
-plan O
-also O
-called O
-the O
-linear B-Math
-hull I-Math
-or O
-just O
-span O
-of O
-a O
-set O
-s O
-of O
-vectors O
-from O
-a O
-vector O
-space O
-denoted O
-spans O
-is O
-defined O
-as O
-the O
-set O
-of O
-all O
-linear O
-combinations O
-of O
-the O
-vectors O
-in O
-s O
-for O
-example O
-two O
-linearly O
-independent O
-vectors O
-span O
-a O
-plan O
-
-however O
-many O
-of O
-the O
-principles O
-are O
-also O
-valid O
-for O
-infinitedimensional O
-vector B-Math
-spaces I-Math
-
-also O
-functional B-Math
-analysis I-Math
-a O
-branch O
-of O
-mathematical O
-analysis O
-may O
-be O
-viewed O
-as O
-the O
-application O
-of O
-linear O
-algebra O
-to O
-function O
-space O
-a O
-branch O
-of O
-mathematical O
-analysis O
-may O
-be O
-viewed O
-as O
-the O
-application O
-of O
-linear B-Math
-algebra I-Math
-to O
-function O
-space O
-
-in O
-mathematics O
-physics O
-and O
-engineering O
-a O
-euclidean O
-vector O
-or O
-simply O
-a O
-vector O
-sometimes O
-called O
-a O
-geometric B-Math
-vector I-Math
-or O
-spatial O
-vector O
-is O
-a O
-geometric O
-object O
-that O
-has O
-magnitude O
-or O
-length O
-and O
-directio O
-or O
-spatial O
-vector O
-is O
-a O
-geometric O
-object O
-that O
-has O
-magnitude B-Attributes
-or O
-length O
-and O
-directio O
-or O
-length B-Attributes
-and O
-directio O
-and O
-direction B-Attributes
-
-here O
-in O
-general O
-means O
-that O
-a O
-different O
-behavior O
-may O
-occur O
-for O
-specific B-Math
-values I-Math
-of O
-the O
-coefficients O
-of O
-the O
-equation O
-
-multilinear B-Math
-maps I-Math
-t O
-vn O
-f O
-can O
-be O
-described O
-via O
-tensor O
-products O
-of O
-elements O
-of O
-t O
-vn O
-f O
-can O
-be O
-described O
-via O
-tensor B-Math
-products O
-of O
-elements O
-of O
-
-a O
-vector B-Math
-is O
-what O
-is O
-needed O
-to O
-carry O
-the O
-point O
-a O
-to O
-the O
-point O
-b O
-the O
-latin O
-word O
-vector O
-means O
-carrie O
-
-the O
-application O
-of O
-linear B-Math
-algebra I-Math
-in O
-this O
-context O
-is O
-vital O
-for O
-the O
-design O
-and O
-operation O
-of O
-modern O
-power O
-systems O
-including O
-renewable O
-energy O
-sources O
-and O
-smart O
-grid O
-
-one O
-may O
-thus O
-replace O
-the O
-field O
-of O
-scalars O
-by O
-a O
-ring B-Math
-r O
-and O
-this O
-gives O
-the O
-structure O
-called O
-a O
-module O
-over O
-r O
-or O
-rmodul O
-
-when O
-the O
-scalar O
-field O
-is O
-the O
-real O
-numbers O
-the O
-vector B-Math
-space I-Math
-is O
-called O
-a O
-real O
-vector O
-space O
-and O
-when O
-the O
-scalar O
-field O
-is O
-the O
-complex O
-numbers O
-the O
-vector O
-space O
-is O
-called O
-a O
-complex O
-vector O
-spac O
-is O
-called O
-a O
-real O
-vector O
-space O
-and O
-when O
-the O
-scalar O
-field O
-is O
-the O
-complex O
-numbers O
-the O
-vector O
-space O
-is O
-called O
-a O
-complex B-Math
-vector I-Math
-space I-Math
-real B-Math
-vector I-Math
-space I-Math
-and O
-when O
-the O
-scalar O
-field O
-is O
-the O
-complex O
-numbers O
-the O
-vector O
-space O
-is O
-called O
-a O
-complex O
-vector O
-spac O
-
-computational O
-algorithms O
-for O
-finding O
-the O
-solutions O
-are O
-an O
-important O
-part O
-of O
-numerical O
-linear B-Math
-algebra I-Math
-and O
-play O
-a O
-prominent O
-role O
-in O
-engineering O
-physics O
-chemistry O
-computer O
-science O
-and O
-economic O
-solutions B-Math
-are O
-an O
-important O
-part O
-of O
-numerical O
-linear O
-algebra O
-and O
-play O
-a O
-prominent O
-role O
-in O
-engineering O
-physics O
-chemistry O
-computer O
-science O
-and O
-economic O
-
-in O
-the O
-first O
-case O
-the O
-dimension B-Math
-of O
-the O
-solution O
-set O
-is O
-in O
-general O
-equal O
-to O
-n O
-m O
-where O
-n O
-is O
-the O
-number O
-of O
-variables O
-and O
-m O
-is O
-the O
-number O
-of O
-equation O
-of O
-the O
-solution B-Math
-set I-Math
-is O
-in O
-general O
-equal O
-to O
-n O
-m O
-where O
-n O
-is O
-the O
-number O
-of O
-variables O
-and O
-m O
-is O
-the O
-number O
-of O
-equation O
-is O
-in O
-general O
-equal O
-to O
-n O
-m O
-where O
-n O
-is O
-the O
-number O
-of O
-variables O
-and O
-m O
-is O
-the O
-number O
-of O
-equations B-Math
-variables B-Math
-and O
-m O
-is O
-the O
-number O
-of O
-equation O
-
-in O
-the O
-case O
-of O
-linear B-Math
-differential I-Math
-equations I-Math
-this O
-means O
-that O
-there O
-are O
-no O
-constant O
-term O
-
-furthermore O
-linear B-Math
-algebra I-Math
-plays O
-a O
-crucial O
-role O
-in O
-thermal O
-energy O
-systems O
-particularly O
-in O
-power O
-systems O
-analysi O
-plays O
-a O
-crucial O
-role O
-in O
-thermal B-Attributes
-energy I-Attributes
-systems I-Attributes
-particularly O
-in O
-power O
-systems O
-analysi O
-
-vector B-Math
-spaces I-Math
-that O
-are O
-not O
-finite O
-dimensional O
-often O
-require O
-additional O
-structure O
-to O
-be O
-tractabl O
-
-the O
-historical O
-roots O
-of O
-functional B-Math
-analysis I-Math
-lie O
-in O
-the O
-study O
-of O
-spaces O
-of O
-functions O
-and O
-the O
-formulation O
-of O
-properties O
-of O
-transformations O
-of O
-functions O
-such O
-as O
-the O
-fourier O
-transform O
-as O
-transformations O
-defining O
-for O
-example O
-continuous O
-or O
-unitary O
-operators O
-between O
-function O
-space O
-lie O
-in O
-the O
-study O
-of O
-spaces O
-of O
-functions O
-and O
-the O
-formulation O
-of O
-properties O
-of O
-transformations O
-of O
-functions O
-such O
-as O
-the O
-fourier B-Math
-transform I-Math
-as O
-transformations O
-defining O
-for O
-example O
-continuous O
-or O
-unitary O
-operators O
-between O
-function O
-space O
-as O
-transformations O
-defining O
-for O
-example O
-continuous O
-or O
-unitary B-Math
-operators I-Math
-between O
-function O
-space O
-
-in O
-general O
-there O
-is O
-not O
-such O
-a O
-complete O
-classification O
-for O
-modules B-Math
-even O
-if O
-one O
-restricts O
-oneself O
-to O
-finitely O
-generated O
-module O
-
-there O
-is O
-a O
-strong O
-relationship O
-between O
-linear B-Math
-algebra I-Math
-and O
-geometry O
-which O
-started O
-with O
-the O
-introduction O
-by O
-rené O
-descartes O
-in O
-of O
-cartesian O
-coordinate O
-and O
-geometry B-Math
-which O
-started O
-with O
-the O
-introduction O
-by O
-rené O
-descartes O
-in O
-of O
-cartesian O
-coordinate O
-
-a O
-linear B-Math
-endomorphism I-Math
-is O
-a O
-linear O
-map O
-that O
-maps O
-a O
-vector O
-space O
-v O
-to O
-itsel O
-is O
-a O
-linear B-Math
-map I-Math
-that O
-maps O
-a O
-vector O
-space O
-v O
-to O
-itsel O
-that O
-maps O
-a O
-vector B-Math
-space I-Math
-v O
-to O
-itsel O
-
-the O
-determinant B-Math
-of O
-a O
-square O
-matrix O
-is O
-a O
-number O
-associated O
-to O
-the O
-matrix O
-which O
-is O
-fundamental O
-for O
-the O
-study O
-of O
-a O
-square O
-matrix O
-for O
-example O
-a O
-square O
-matrix O
-is O
-invertible O
-if O
-and O
-only O
-if O
-it O
-has O
-a O
-nonzero O
-determinant O
-and O
-the O
-eigenvalues O
-of O
-a O
-square O
-matrix O
-are O
-the O
-roots O
-of O
-a O
-polynomial O
-determinan O
-of O
-a O
-square B-Math
-matrix I-Math
-is O
-a O
-number O
-associated O
-to O
-the O
-matrix O
-which O
-is O
-fundamental O
-for O
-the O
-study O
-of O
-a O
-square O
-matrix O
-for O
-example O
-a O
-square O
-matrix O
-is O
-invertible O
-if O
-and O
-only O
-if O
-it O
-has O
-a O
-nonzero O
-determinant O
-and O
-the O
-eigenvalues O
-of O
-a O
-square O
-matrix O
-are O
-the O
-roots O
-of O
-a O
-polynomial O
-determinan O
-is O
-a O
-number O
-associated O
-to O
-the O
-matrix O
-which O
-is O
-fundamental O
-for O
-the O
-study O
-of O
-a O
-square O
-matrix O
-for O
-example O
-a O
-square O
-matrix O
-is O
-invertible O
-if O
-and O
-only O
-if O
-it O
-has O
-a O
-nonzero O
-determinant O
-and O
-the O
-eigenvalues B-Math
-of O
-a O
-square O
-matrix O
-are O
-the O
-roots O
-of O
-a O
-polynomial O
-determinan O
-of O
-a O
-square B-Math
-matrix I-Math
-are O
-the O
-roots O
-of O
-a O
-polynomial O
-determinan O
-are O
-the O
-roots O
-of O
-a O
-polynomial B-Math
-determinant I-Math
-
-in O
-this O
-context O
-the O
-elements O
-of O
-v O
-are O
-commonly O
-called O
-vectors B-Math
-and O
-the O
-elements O
-of O
-f O
-are O
-called O
-scalar O
-and O
-the O
-elements O
-of O
-f O
-are O
-called O
-scalars B-Math
-
-matrices B-Math
-are O
-used O
-to O
-represent O
-linear O
-maps O
-and O
-allow O
-explicit O
-computations O
-in O
-linear O
-algebr O
-are O
-used O
-to O
-represent O
-linear B-Math
-maps I-Math
-and O
-allow O
-explicit O
-computations O
-in O
-linear O
-algebr O
-and O
-allow O
-explicit O
-computations O
-in O
-linear B-Math
-algebra I-Math
-
-such O
-a O
-system O
-is O
-also O
-known O
-as O
-an O
-overdetermined B-Math
-system I-Math
-
-because O
-an O
-isomorphism B-Math
-preserves O
-linear O
-structure O
-two O
-isomorphic O
-vector O
-spaces O
-are O
-essentially O
-the O
-same O
-from O
-the O
-linear O
-algebra O
-point O
-of O
-view O
-in O
-the O
-sense O
-that O
-they O
-can O
-not O
-be O
-distinguished O
-by O
-using O
-vector O
-space O
-propertie O
-preserves O
-linear O
-structure O
-two O
-isomorphic O
-vector O
-spaces O
-are O
-essentially O
-the O
-same O
-from O
-the O
-linear B-Math
-algebra I-Math
-point O
-of O
-view O
-in O
-the O
-sense O
-that O
-they O
-can O
-not O
-be O
-distinguished O
-by O
-using O
-vector O
-space O
-propertie O
-isomorphic B-Math
-vector I-Math
-spaces I-Math
-are O
-essentially O
-the O
-same O
-from O
-the O
-linear O
-algebra O
-point O
-of O
-view O
-in O
-the O
-sense O
-that O
-they O
-can O
-not O
-be O
-distinguished O
-by O
-using O
-vector O
-space O
-propertie O
-
-sometimes O
-the O
-term O
-linear O
-function O
-has O
-the O
-same O
-meaning O
-as O
-linear B-Math
-map I-Math
-while O
-in O
-analysis O
-it O
-does O
-no O
-linear B-Math
-function I-Math
-has O
-the O
-same O
-meaning O
-as O
-linear O
-map O
-while O
-in O
-analysis O
-it O
-does O
-no O
-
-infinitedimensional O
-vector B-Math
-spaces I-Math
-occur O
-in O
-many O
-areas O
-of O
-mathematic O
-occur O
-in O
-many O
-areas O
-of O
-mathematics B-Math
-infinitedimensional B-Attributes
-vector O
-spaces O
-occur O
-in O
-many O
-areas O
-of O
-mathematic O
-
-in O
-particular O
-over O
-a O
-principal O
-ideal O
-domain O
-every O
-submodule B-Math
-of O
-a O
-free O
-module O
-is O
-free O
-and O
-the O
-fundamental O
-theorem O
-of O
-finitely O
-generated O
-abelian O
-groups O
-may O
-be O
-extended O
-straightforwardly O
-to O
-finitely O
-generated O
-modules O
-over O
-a O
-principal O
-rin O
-of O
-a O
-free O
-module B-Math
-is O
-free O
-and O
-the O
-fundamental O
-theorem O
-of O
-finitely O
-generated O
-abelian O
-groups O
-may O
-be O
-extended O
-straightforwardly O
-to O
-finitely O
-generated O
-modules O
-over O
-a O
-principal O
-rin O
-
-module O
-homomorphisms O
-between O
-finitely O
-generated O
-free O
-modules O
-may O
-be O
-represented O
-by O
-matrices B-Math