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1603623259 Aghalibrary.Pdf Aghalibrary.com Aghalibrary.com Aghalibrary.com ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra "ر! $ط ا%ر ن ړۍ & د ر* وو/م ز.ر-د دی ، واړم $ط ا %ر -0 و*و2ت & ز ت 2وت ري او ر ر*0 "$ ورو *و & ر ی ھم ډره ز ا5ده ږي ، د را6ول او رر & راوړم . ددي 8 %7 د %وور"9 د ور5و :وو-0 8رو ، & ا!:05 اور0 ژ%ود$ط0 ا%ري %ر$ 0 ور.ړل >ودي ، مھ >ل دی . آرزو رم & 5ب و$ت & دي @ل ډول و5?ت ور ړم ر "و $ط ا%ر د ز& ا5دي وړ و.ر-ول >0 . $ط ا %ر رو5م ړۍ وري د و A5ل *ون >8ل وود و . %:8 و 5ت د :: ھد5 و . د Bل ډول Leibniz 1690 ل & و Cرول د و ر8س د در ت 5E راو5:و ره دا ړ . !ر ن $ط ا %ر وازی وه / %ر$ د ر*و .ر-د0 ده ، %:8 $ط ا %ر "$ د 55 وا ا @دی 5:و ل & ھم ری ډره ا5ده ږي . د Fو Hښ >وی دی & ھI ر*0 5%و و ا5?ل >&، & ن ړۍ & روج او %و & ھI "$ ا5ده ږي او 7% د وي ?0 ژ%& وي . ھ دار! د ووو ا5?ل & وHښ >وی دی & د دو ژ%و ( Hو ا و اK:5 ) رو ووو "$ ا5ده و>0 . ا% ھI ھم 0 وږ $:و:0 ژ%و 0 ھI ووا د او5 درد وم رو، ا!:5 8ل >ودي . دا دھIو 5 وره & د ر* اوور 2و :و % و %ن ا::0 ژ%و ط? وي ، د .& وړ% ھم وي . ھI 5 %و و ا و ا$@رات & د ا5?ل >وی دي ، ا$ر & & >رL ړ ی دی. د DAAD اM و55 "$ >8ر وم & د درس و 7 را د !رھراوھرات د وھوو د 5س وھ-و & 25ده ړي وه . ھدار! د Afghanic و505 "$ وم وو را5ره ر5 ړده . ا 8ن ری & 8:وا و د N و وړHت & ا>%/ت :طM وود ي وي ، ?ذرت واړم . "ر.ده ده ھ ر0:2 اBر 8ل ږي -0 !ړ:وي رې، ود روو5و8و "$ ھ: وم & دا !ړوي د$ل وړادز ډول ز Eدي ا8رو8& & راو5وي، ر "و را:و08 & م & وول >0 . درHت [email protected] ډ ارد د 2 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra رت ړی ل ( &روع $#" 6 ): ): ر ' ات : و" (set) : : ?ن 5ت (finite set) ، ر?ن 5ت (infinite set) ، countible set ( د >ر وړ 5ت ) ، infinite countible set ( د >رش وړ ر?ن 5ت ) ، uncountible set : ( mapping) ) ا8ب (injective) ، 5ور8ف (surjective)، %8ف (bijective) ار (algebra) : : دو .و0 را%ط (binary operation) ، ا%ری وړHت ( 5$ن ) (algebraic structure) ، . روپ (field) 5 ، (ring) A: ، (group) راط" (relation) : : ا?8س را%ط ( reflexive ) ، ظره را%ط (symmetric) ، اtransitive) A) را%ط ، ?د را%ط ( equivalence relation )، ر%وط ا55 *وي دو م ل ( &روع $#" 31 ): ): 31 د ط ,د+و م ( System of linear Equation ): ): د $ط 5و (homgen) ?دEو او ر5و (inhomogen) ل ، Q وس طرGaussian Algorithm ) A )، ر%وط ا55 *وي در م ل ( &روع $#" 43 ): ): ر س ا و د ر ت ( Matrix and Determinant): ر8س ، درت ، د ?8وس ر8س ددا ووطر0A ، ل 55م د $ط ?دE و ل د ر8س وا5ط ، د cramer طرA ، ور (minor) ، وCور (cofactor) ر8س /.ورم ل ( &روع $#" 75 ): ): ووری ' (Vectorspace): و وری C ، *Cرsubspace) *C 2 )، $ط ر ب ( ) ، span ، Linear Combination 3 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra $ط واLinearly dependent ) 5% ) ، $ط A5ل Linearly independent ) ، ر%وط ا55 *وي 01م ل ( &روع $#" 92 ): ): د 'ی وور 2 ده ا و ,د : : ( basis and dimension of a vectorspace ) 2ده (basis) دوی و وری C* ، ا55 2ده ( canonical basis )، 2ده (basis) د وی Cرsubspace) *C 2) ، د ر8س رrank) R) ، %?د (dimension) دوی و وری C* ، ر%وط ا55 *وي &1ږم ل ( &روع $#" 118 ): د ر 'و و" (sum of subspaces) : : %?د Cرول د CرC 2* .و ( Dimension Formel for subspaces ) د CرC 2* .و A5 وdirect sum of subspaces) 2 ) ، ر%وط ا55 *وی اوم ل ( &روع $#" 127 ): ط 1 5 ( 7ش ) ( linear mapping :) :) ھوورCزم (homomorphism) ، ووورCزم ( Monomorphism ) ، اوورCزم ( epimorphism ) ، ازوورCزم ( ismorphism ) ، ادوورCزم ( Endomorphism ) ، اووورCزم ( Automorphism ) Image او د و $ط kernel S ، %?د Cرول د $ط Dimension Formel for linear mapping ) 9 ) ، اوارت ( C ( InvariantرC 2* ام ل ( &روع $#" 154 ): د ر س ا وط 51 ر 8 راط" : : (Linear Mapping and Matrix ) د$ط 9 ر%وط ر8س ظرا55 2دي د ر8س ر%وط $ط 9 ظرا55 2دي را%ط د$ط ) 9Aش ) اور8س رT ظر دو $:و 2دو ، skew Hermitain matrix ، Hermitain matrix ، adjiont matrix ، idempotent matrix ، nilpotent matrix ، involutory matrix ، permutation matrix ، ر%وط ا55 *وي م ل ( &رو ع $#" 176 ): & " 2 و" او & " وور و" : : 4 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra ( Eigenvalues and Eigenvectors ) >$@ و ور و ( eigenvectors ) >$@ و ( eigenvalues) ( eigenspace ) *C @$< (characteristic function ) 7% @$< ھد5 @ل *رب ( geometric multiplicity ) ا%ری @ل *رب ( algebraic multiplicity ) ) ارو Qو ل ( orthogonal ) ر8س ، د.ول ( diagonal ) ر8س ، diagonalizable ر8س ، ?دل ( equivalence) ر8س، >% ( similar) ر8س ، upper triangular matrix ( ور B:B ر8س ) ، ر%وط ا55 *وی م ل ( &روع $#" 199 ): ا2. دی ' (euclidean space) : : Bilinearform ، 85ری @ل *رب ( scalar product )، ا :دی normed vector space ، norm ، ( euclidean space ) * C ، رR و وری ortogonal ، ( metric space ) *C و ورو ، orthonormal و ورو ، اوروورل 2ده ( orthonormalbasis ) ، gram-schmidt process ، و وری @ل *رب ( vectorproduct)، hermitian ، unitary vector space ، semi-bilinear ، ر%وط ا55 *وي وم ل ( &روع $#" 214 ): د ر و ډوو" او ا,ل : : Quadratic Form ( دو در Cورم او ر%? Cورم ) ) negative definite ، positive definite ، positive semidefinite principal minor ، indefinite ، negative semidefinite ، Hessian Matrix ، jacobian matrix ( ھس ر8س ) ) local maximum ( و*? ا2ظ او 5% ا2ظ ) ) local minimum ( ا@Iری و*? ا@Iری %5 ) ) wronskian matrix ، Cayley-Hamilton theorem ر%وط ا55 *وي دو م ل ( &روع $#" 237 ): 9و" او ر و" 5 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra ړی ل ( و" , ا0ور( و ر) او اړ ( راط") ) ( Set , Mapping and Relation ) دې C@ل H واړم - ھم او *وی & وور 5 $ط ا %ری H ور - ا5ده ږی $@رډول >رL ړم . ,ر ف 1.1: 5ت ( set ) د Georg Cantor $وا 1874Nدي ل & Eدې ډول ?ر ف >وی دی : : Set وه و2 د او%8وو ( Objects ) ده & 6ول و ?ن >$@ت وری !ر و %ل "$ Cرق ري . دBل ډول X 5ت 5س د وھ- @:ن وي . ?ن >$@ت د د5س دوھ- @ل دل دي . !رھر @ل و%ل "$ Cرق ری. وږ و 5ت Eدی ډ ول Hو : : ͒ = {ͬͥ, ͬͦ,………….} د Objects يد & د X د 5ت د 2@رو cardinality X . (elements) ͥͬ ,ͦͬ ,ͧͬ . وم . دږی دوه 5ت د 2@ر و>رد وم د ږي اووږ ھI 5ره Hو . $ 5ت 5ره Hودل ږي. : : 1. 2 , ر ف |͒| ∅ ( a ) ری X او Y دوه 5 وی . C Xر2 5ت (subset) د Y ول ږی ېد >رط & : : (͒ ⊆ ͓) و Cر2 5ت (proper ⟹ x ∈ subset) Y X X د ∋ x Y ∀ ول ږي ېد . X Y >رط & H - 2@ر وود وي &( ⊂ Y X) & >ل وی a ; a X :? د Bل ډول∌ Y ∋ ∃ X = { 2,4,5}, Y = {2,4,5,a,b} ھر5ت و$ ͓ ⊃ Cر2 5͒ت ⟹ری . X او Y 5ره 5وی دی ېد >رط & او وی . ?: ͓ ⊆ ͒ ͒ ⊆ ͓ 6 ͒ = ͓ ⇔ (͒ ⊆ ͓) ∧ (͓ ⊆ ͒) ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra ( b ) د?ن 5ت ( finite set) ره ډول ډول ?ر و و ود دي R . Dedekind (1916-1831) ?ن 5 ت ( finite set) دار ! ?رف ړی : و5ت X ?ن ول ږي دې >رط & X ھU و Cر2 5ت (proper subset) وود وې & Cardinality (د 2@ر و >ر ) د X 5ره 5وی وی . ? A X ; اودا& : |͒| = |̻| ⊃ ∄ A X ; | | | | وږ ?ن 5ت ͒ > ̻ ⊃5ره Hو . ∀ د د X د 2@ر و >ر 5وی ∞ n دی{ . (ͬ , ? … , ͦͬ ,ͥͬ} = ͒ . (infinite set) ھر ھI 5ت ≠ & ?ن و ي د͢ = |͒|ر?ن 5ت وم دږي ∞ : ? : : 9ل = |͒| ℕ = { 1,2,3 ,3,4,…} ℕͤ = { 0,1,2,3 ,3,4,…} ℤ = { ...,−3,−2,−1 ,0,1,2,3 ,…} 2ℤ = { ...,−6,−4,−2 ,0,2,4,6 ,…}+ ℚ = {, │ͤ,ͥ ∈ ℤ,ͥ ≠ 0} : : . ور 5 و 6ول ر?نℂ يد⊇ ℕ ⊆ ℤ ⊆ ℚ 8- ⊆ ℝ ( ( ℕ ⊂ ℤ∞ ⊂ ℚ ⊂ ℝ∞ ⊂ ℂ , 2ℤ ∞ ⊂ ℤ ) ∧∞ ∞ = ∞ ) |ℕ| = , |ℤ| = , |2ℤ| = , |ℚ| = , |ℝ| = , |ℂ| 9ل : د اEدی 5و ?ن دي و % ر دی ͒ = ƣ ͬ ɳ │ ͬƧ X ا و Y ھر و 5 {2 ≥ 2@ر ه ͭ ≥ ری − . 2? ℤ ∋ ͭ } = ͓ !ر او 5 = |͓| = | ͒| ͒ ⊈ ͓ ͓ ⊈ ͒ ͑ ͥ : = {ͫ ∈ ℤ |−15 ≤ ͫ ≤ 16 } 7 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra ت ͑ͦ: = ƣͫ ∈ ℤ ɳ1 ≤ ͫ ≤ 16 ∧ (͙͙ͪ͢) ͫ Ƨ دل ږی & ا و {6,8,10,12,14,16 ,2,4} = دا Eدی 5 و ͥ $͑ دی⊇ ͦ͑ 8 = |ͦ͑| , ͦ ͧ { | } { | } ا و = 0 x = 3 = ͬ ∈ ℤ W4: n < 0 ℕ ∋ ͢ = : ͑ | ͧ| | ͨ| | ͩ| ,ر ف 1.3 : ͑ =ری͑ = ͑ 5 و وي : : ͒ͥ, ͒ͦ, … , ͒) اﺗﺤﺎد : (±Ä¿ÅÄ ) ͒ͥ ∪ ͒ͦ ∪ … ∪ ͒) ≔ {ͬ | ∃͝ ∈ {1,2,3, . , ͢}; ͬ ∈ ͒$} Aط7 : (¿ÄÊ»ÈÉ»¹Ê¿ÅÄ ) ͒ͥ ∩ ͒ͦ ∩ … ∩ ͒) ≔ ƣͬ ɳ ͬ ∈ ͒$, ∀͝ ∈ {1,2, … , ͢}Ƨ ور Bل = ا و = دی 9ل : 5ت ͥ͑دھA ͑ͦ IA ∪ 2͑ͥددو وي ͦ &͑ د@ͦ͑ ∩ر"$ ͥ͑زت او 5وی دی او 5ͮتℝ دھA IA 2ددو وي & د@ ر"$ م او 5وی دی . ? ℝͯ = ͮ ℝ = {ͬ ∈ ℝ | ͬ ≥ 0 } دھIوی ا د د AA ا2داد و 5ت وا {دھIوی A ≤ 0ط 7 ͬ | ℝ @ر ∋ دیͬ .} ℝͯ ? = وا {0} = 9ل : ℝͯ ℝ ∪ ℝͯ ℝͮ ∩ ℝͮ │ ͒ ≔ {x ∈ ℤ │ (−8 ≤ ͬ ≤ 8)} 8 8 Y ͓ ≔ {x ∈ ℤ (−8 < ͬ < 8)} -8∈ X ⟹ -8 ∈ Y X ∪ -8 X Y 5 ∈ X ∧ 5 ∉ ⟹ 5∉ X∩Y 9ل : ∩ ∋ ⟹ X ∧ ∈ Y ∋ 8 ﺧﻄﻲ اﻟﺠﺒﺮ ---------------------------------------------------------------------- Linear Algebra A = {a,b,c,d} , B = {d,e,f}, C = {a,b } A B = {a,b,c,d,e,f} , A B = {d} C ∪ A , A C = {a,b,c,d} ∩ = A , A C = {a,b} = C ⊆ ∪ ∩ A B = { a A │ a B } = {a,b,c} ∖ ∈ ∉ A C = { a A │ a C } = { c, d } , C A = "ر ! & C A ∅دی .
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