. ) ( :

: 1 : ................................ ................................ ................................ ................................ .................... 21 - 1 - ................................ ................................ ................................ ................................ ................................ ...................... 21 - 2 - ................................ ................................ ................................ ................................ ................................ ................................ ......... 21 - 3 - ................................ ................................ ................................ ................................ ................................ ................................ ...... 2 ................................ ................................ ................................ ................................ ................................ .......................... 3 ) Inverse ( ................................ ................................ ................................ ................................ .............................. 5 2 : ) Field, Vectors and Spaces Vector ( ................................ ................................ .......... 72 - 1 - ) Field (................................ ................................ ................................ ................................ ................................ .............................. 7 3 : ................................ ................................ ................................ ................................ ......... 113 - 1 - ) Orthogonal Vectors ( ) Orthonormal Vectors :( ................................ ........... 12 4 : ) Mapping (................................ ................................ ................................ ....................... 13 5 : : Eigen Value / Eigen Vector ................................ ................................ ................................ 17 6 : ) Linear Functional :( ................................ ................................ ................................ ................................ ........ 23

zeinali80@gmail.com . 2 . 1 : 1 - 1 - - : .] 2 [ - : - : 1 - 2 - : .1 - A B, B C A C2 - A B, B A A = B3 - ) Union ( ) Intersection ( .S1 S2 { o: o S1 o S2} S1 S2 { o: o S1 o S2} 4 - A AC { o: o A} .5 - ) Associative Law :( ( A B) C = A ( B C) ( A B) C = A ( B C)6 - ) Commutative Law :( A B = B AA B = B A7 - ) DistrbutiveLaw :( A ( B C) = ( A B) ( A C) A ( B C) =( A B) ( A C)8 - ) Identity Law :( A 1 = AA 1 = 1A u = uA u = A u 9 - : Ac A = uAc A = 110 - ) DeMorgan's Theorem :( ( A B)c = Ac Bc( A B)c = Ac Bc : A B = { ( o, b) ; o A, b B} ( o, b) .Rn = R R = |( o1, , on) ; o] R| Rn [ o1, , on]1 .1 - 3 - Am n |o]]m n 3 : |o]]m n = |b]]m n : |o]]m n |b]]m n = |o] b]]m n : |oo]]m n : : |o]]m n |b]]n p = |c]]m p = okbk]nk=1; i = 1, , m; ] =1, , p 1 - : A + B = B + AAB BA2 - : A( B + C) = AB + AC3 - : A + ( B + C) = ( A + B) + CA( BC) =( AB) CAB = I = BABI = B = IB Transpose : An m1= |o]] : ( ABC)1 = 1C1B1A1 Symmetric : : A = A1) ( Skew Symmetric : A = A1 : A AA1 A1A A + A1 A A1 Conjugate : A |o]] : A = A : A = A Associate : AH= A1 |o]] Hermitian : : A1= A : : A1= A : : AA1= A1A : : A = A : : ) Minor ( Mpq: n2 4 ) Cofactor ( Cpq: opq Cpq Cpq = ( 1)p+qHpq . ) Determinant :( . : : k : | A| = ok]Ck]n]=1 : k : | A| = okCkn]=1 : : : - 2 2 o bc J = oJ bc - 3 3 _o b cJ c g i_o bJ cg oci + bg + cJ ( bJi + o + ccg) - _ 1 1 1x y zx2y2z2_ = ( y x) ( z x) ( z y) _1 + x y zx 1 + y zx y 1 + z_ = 1 + x + y + z :1 - .| ABC| = | A| | B| | C|2 - | A1| = | A|3 - 4 - ) ( 5 - ) ( ) ( 6 - ) ( 7 - o on :| oA| = on| A|8 - ) ( o ) ( ) ( 59 - 10 - |Am pBp n| = 0 i p < m 11 - | Au| = | A|u 12 - ) ( ] C ) ( ] A + B : | C| = | A| + | B| : 21 31 32 41 42 43 : _o b cJ c g i_ J g c b ) ) Adjoint Matrix ( CT: C A ) Inverse ( A-1 =CT| A| AA-1 = I = A-1A ) Singular Matrix :( ) ( ) NonSingularMatrix :( ) ( : : A B AB A1 .o-1 =1oI-1 = I( A-1)n = A-n( ABC)-1 = C-1B-1A-1| A-1| =1|A-1|( A1)-1 =( A-1)1i A1= A-1 | A| = 1jo bc J[-1=1cd-bcj J bc o [ _o 0 00 b 00 0 c_-1= _1c0 001b00 01c_ _0 0 c0 b 0o 0 0_-1= _0 01c01b01c0 0_ : : : 6 ) ( ) rank ( rA: A ) ( rA min{ m, n} : ) ( : o : = 0 o ) o : = 0 ( o o = min{ m, n} C = AB rC =min{ rA, rB} ) Trace ( Tr( A) : ) Ir( A) = o( Ir( AB) = Ir( BA) Ir( A + B) = Ir( A) +Ir( B) Ir( oA) = oIr( A) . : A . : A = _A1A2A3A4_ = _A5A6_ = [ A7A8A9] = : 0 A = 0 A B A B A p q B q r . : AB] B]A : 0 A = 0 B A B A A B A B _I100 I2_ = I B . 7( )( )1112 3 4 2 1 11 2 1 2 11 2 4 21 113 4 3 4 23 4 3 14 4 3 1 20;0B A A A A IB B A A IB B A AB B A A IB B A AB A A A A = ((( = = ((( = = A ._A100 A2_-1= _A1-100 A2-1_ _A100 A2_ = | A1| | A2| ) Mitary ( orthogonal : : A1A = I = AA1 ) A1= A-1( : A B AB A1 A-1 2 : ) Vector s and Spaces Vector Field,( .2 - 1 - ) Field ( ): (1 - .2 - o, b ( o + b) 3 - o, b ( ob) 4 - o 0 + o = 0; 0 o = 05 - : o, b ; b 0 ( o/ b) 6 - o ( o) ; o + ( o) = 07 - : 6 . z : x n x = [x1 xn]1 Rn Cn . ) InnerProduct :( u : = u1: = u1:1 ++ un:n = |u||:| cos 0 : [u[ u u = o12+ + on2 : ) u : = u :1( . 8 : I .) (1 - 2 - :1, :2 I :1 + :2 I3 - :1, :2 I :1:2 I 4 - :1 I 0 + :1 = 0; 0:1 =0 5 - :1 I ( :1) I: :1 + ( :1) I6 - .7 - :1, :2 I o:1 I; [:2 I; ( o:1 + [:2) I :1 - Rn R 2 - Cn C 3 - 4 - n 5 - . :1 R2 R3 .2 - Rn 3 - Rn : { x} ) (1 - n n ) ) ( ( : 4 3 2 2 2 2 2 9 : m n : ) :( ) ( ) (2 - m n ox = 0 o= 0 : ) ( : o: = 0 o ) o: = 0 ( o o = min{ m, n} 3 - m n : i | 0| = 0 x = [x1 xn]0m m = _x1 x1 x1 xm xm x1 xm xm_5 - : ) ( : : n n Rn ) x = [x1 xn] x1 = = xn ( : . 10 : n 1 1, t, t2, t3, , tn-1 [ 1,0, ,0]1, , [ 0, ,0, tn-1]1 : w :1, :2, :3, :4 . :1 = 2 + t + t2 + 4t3; :2 =2 + t + 2t2 + t3; :3 = 1 + t + t2 + t3; :4 = 3 + 2t + 2t2 + 5t3; ._2 1 1 42 1 2 11 1 1 13 2 2 5_ _2 1 1 40 0 1 30 1 1 20 1/ 3 1/ 3 2/ 3_ _2 1 1 40 0 1 30 0 0 00 1/ 3 1/ 3 2/ 3_ w : :1 = t + t2 + 2ct + cost; :2 = ; :3 = o: = ( )=0t + ( )=0t2 + ( )=0ct +( )=0cost [ 1,0,0] , [ 0,1,0] , [ 0,0,1] 1, t, t2, : x + 3xi + 2x = 0 : x y x, y p ox + [y p; 0 p ) Rn Cn ( : xy R3 R3 . z R3 z > 0 R3 R3 ) ( : x1 = [ 1,2,3,4,5]1 x2 = [ 1,0,0,0,0]1 x3 =[ 0,1,1,0,0]1 ) ( R5) 5 ( : : . ow1+ [w2 : n Rn ) ( !! : 11 : R3 ) A, B oA + [B ( : ) ( : : u w u + w ) u w (!! : uw u w y 1 . y 2 . .y = uw y = u + w, u w = { 0} : u = { ( o, b, 0) } xy w = { ( 0, c, J) } yz :u + w = ( o, b, 0)+ ( 0, c, J) = ( o, b + c, J) R3 u + w = R3; u + w R4

u w = { ( 0, y, 0) , c R} 0 uw R3

: u xy w z R3 R3 u w 3 : x, y I ) ( ) ( o = x y :1 - x + y, z = x, z + y, z ; ox, y = o x, y ; ox1 + [x2, y = o x1, y + [x2, y2 - x, y = y, x

3 - x, y 0 x = 0 4 - x, x > 0; i x = 0 x, x = 0 : 1 . ox1 + [x2, y = o x1, y + [x2, y 2 . x, y = y, x

3 . x, x > 0; i x = 0 x, x = 0 :1 - Rn: x, y = x1y ) ( 122 - Cn: x, y = x1 y ) ( x1 y = y1x

= y1x3 - : , g = ( z) g( z) Jzbu 4 - : , g = ( z) g( z) Jzbu= Jzbu+ ] Jzbu ) o b = u + ]: u : ) ( : 5 - : Am n, Bn p = Ir(A1 B)3 - 1 - ) OrthogonalVector s ( ) Orthonor mal Vector s :( .:, :] = o]= _0 i ]1 i = ]

: c = [ 0, ,0,1,0, ,0]1 n y : . : = y jikkk:k-1k=1; : =i[i[ : : x _ ) x = o:( o : :1 - o= : x2 - o=ix[i[ 3 - : ) : w ( I : ) w = I-1( w w1 . o= w x ) w w :] = o]( 13 4 : ) M apping ( : x y x y ) ( x y y = ( x) m n 1 . : Am n: Rn Rm; : Rn, w Rm: A: = w2 . : : I I, I = { ontn + on-1tn-1 + } ) : ( 4t) =4; 4t I, 4 I (3 . : I: I R ) : 2tJt10= 1; 2t I, 1 R (4 . : F( x, y) = ( x + 1, y + 2) : Domain : Range : y Image : y : ( x, y, z) = ( x, y, 0) R3 R2 : : x y; g: y z og: x z . o( go) = ogo = ( og) o og go : : = [x1 xn] one-to-one : y x . o1 o2 ( o1) ( o2) ( o1) = ( o2) o1 = o2 On to : : ) y x x y y x : o, [ ; :1, :2 I; F: I : F( o:1 + [:2) =oF( :1) + [F( :2) 0 . . : . ! .Am n: Rn Rm; : Rn: A(o:1 + [:2) = A (o:1 + [:2) = oA :1 + [A :2 =oA(:1) + [A(:2) ) I: F F :( ) : I: I( :) = : ( 14 kernel : ) ( ) ( Ic: = 0 F: I R: kcr( F) = { : I: F( :) = 0} ) Ax = 0 ( F( :) = 0 ) ( Jim( kcr( F) ) : ) F: I u ( ) u ( ) I ( : . Jim( kcr( F) ) +Jim( Im( F) ) = Jim( I) : I2( u) = I( I( u) ) Operator : ) (I: I I Linear Operator : I: I I : |c| = { [ 0, ,1,0, 0]1} I : I :: = oc = o1_10_ + + on_01_ = _1 0 0 1_ _o1on_ = _o1on_ : |c| : F( x, y, z) = ( x + y, y + z) : R3 R2 I|c| F( c1) = ; F( c2) = ; F( c3) = j10[ , j11[ ) I |c| :Ic = I|c1, , cn] ( 1, , n:I( 1) = = o111 + + o1nn = _o11o1n_I( n) = = on11 + + onnn = _on1onn_ I]c= _o11 on1 o1n onn_ . : I : I] I : .I( :) ]c= I]c:]c

15 : I( 1) = o11g1 + + o1ngn = _o11o1n_ I]g= _o11 on1 o1n onn_ g : |c| ) ( : ) (... : Im : F( x, y, z) = ( 3x + 2y 4z, x 5y + 3z) ; g1 = ( 1,3) ; g2 = ( 2,5) ; 1 = ( 1,1,1) ; 2 =( 1,1,0) ; 3 = ( 1,0,0)F( 1) = ( 1, 1) = o1g1 + o2g2 F]g= _o1[1y1o2[2y2_2 3

: ) ( I I ) ( I I |c| ]1 = I( c1) = ? = = o11c1 + + o1ncn = _o11o1n_n = I( cn) = ? = = on1c1 + + onncn = _on1onn_ P = _o11 on1 o1n onn_ P ) ( |c| ] ) |c| ] ( . : . : . : ) ( |c| ] I :] = P-1:c : ) ( |c| ] I I I] = P-1IcP ) P-1 = ( ) ( 16 : I( 1,1) = ( 0,2) ; I( 3,1) = ( 2, 4) I( o, b) = ? . ( o, b) .I = j0 22 4[ I( o, [) = j0 22 4[ jo[[ = _2[2o 4[_ ; ( o, b) = o( 1,1)+ [( 3,1) ; o =3b-c2, [ =c-b2

( o, b) = o( 1, 1) + [( 2,3) ; o = ? , [ = ? : I( o, b) = I(o( 1, 1)+ [( 2,3) ) = oI( 1, 1)+[I( 2,3) = ( o, b) : R3 R4; :1 = [ 2,0, 1, 3] ; :2 = [ 1,2,0, 4] I = ? I( c1) = :1; I( c2) = :2; I( c3) = :3 I( x, y, z) = xI( c1) + yI( c2) + zI( c3) : A B B= P-1AP ) ) ( ( : ) um 1( t) = um 1( t) ; y 1( t) =y 1( t) ( ._xn 1( t) = An nxn 1( t)+ Bn mum 1( t)y 1( t) = C nxn 1( t) + mum 1( t)

xn 1( t) um 1( t) y 1( t) An n Bn m C n x = P-1x; x = Px _x`n 1( t) = A`n nxn 1( t)+ B`n mum 1( t)y 1( t) = C` nxn 1( t) + mum 1( t) A` = P-1AP; B`=P-1B; C` = CP; = : . x = P-1x A` = P-1AP : P-1AP = H-1AH ) : ( I Ic ) ( Ic Image ) ) ( Ic ( :1 :2 ) :1 :2 ( : H .I( A) = AH HA Jim( kcr) = ? I( A2 2) = jx yz t[ H HA = j0 00 0[ 17 5 : : Eigen Value / Eigen Vector I ) I: : : ( z I : I I(:) = z: : z z ) I ( : : I I: x = c5t I ( c5t) = 5c5x = 5x z = 5 : : I : ) ( : ) ( ) ( : A ) ( ) A :( A: = z: or ( A zI) : =0 or | A zI| = 0 : An n n : z ) z z ) o11 (( : z . ] Error! Bookmark not defined. [A: = z: ( A zI) : = 0 ( zI A) : = 0 zI A Resolvant A . ] Error! Bookmark not defined. [ : ) ( A zI ) A: = z: : N( A zI) ( ] Error! Bookmark not defined. [ : A = A1 . ] Error! Bookmark not defined. [ 18 : : A : . A: = z:] Error! Bookmark not defined. [ : . : . : ) ( Ax1 = z1x1 x1 x2 Ax2 = z2x2 x2 x1 ) x1, x2 = 0 ( x3 Ax3 = z3x3 x3 x1 x2 . : ( A) = onAn + on-1An-1 + + o0I ( A) = 0 : I .: : : ( I) = onIn + on-1In-1 + + o0I : An n A zI zI A | A zI| = 0 A : ( z) = | A zI| : ( z) = | A zI| = zn + o1zn-1 + + on-1z + on : | A| = ( 1)non = z Ir( A) = o= o1z : _A100 A2_ ( z) = | A zI| =| A1 zI| | A2 zI| A : ). P P ( A` . @ ] 1 [ 19P = [:1 :n] P A ) ) ( z ( : P . @ ] 1 [ : P ) Modal ( . : o + ]b o ]b P .@ ] 1 [ : z1,2 = o ] :A` = P-1AP = j o o[ : P ) ( . : A1A = AA1 I . @ ] 1 [ : x = P-1x P A :A` = P-1AP = A = Jiog( z1, , zn) = _z1 0 0 zn_ x` = Ax + Bu _x`1 = o1x1 + b1u1( t)

A . x = P-1x A` = P-1AP . : : . ) z ( ) .( . : |A`| = | A| ; Ir(A`) = Ir( A) 20 : A B ) (... B = P-1AP : : A A : A` = H-1AH : A ) ( P-1AP = A A n A zI z 1 . 2 . 3 . : ) Canonicalform ( : n ) ( n 1 . ) Triangle ( 2 . Diagonal3 . .: = [x1 xn]?? A ) ( : Multiplicity :1 - m|: z 2 - q| = n rank( A 2|I) : z n ) .( 1 q m : . : n n H-1AH = g H P ) ( H-1 P-1 ) . ( : ) z: m= q( m q n ) ( n H-1AH = g 21 : ) q= 1 ( m q=m 1 z H ) ( H-1AH A ) [ ( : ( A zI) : = 0 : _( A zI) :g1 = :( A zI) :g2 = :g1 12121230 0 0 01 0 00 1 0 000 1 00 0 1 0;00 0 10 0 0 00 0 00 0 0 0iiiiiiJJ JJ ( ( ( ( ( ( ((= = = ( (( (( ( : ) 1< q < m ( m q H H-1AH ) [ ( ) ( 3 3; 1 1 2 2; 2 2 : _( A zI) :g1 = :1( A zI) :g2 = :2 :21 11 12 12 1311iiJJJJJ ( ( ( ( ( ( (= =( ( ( ( ( ( ( ( 22 H-1AH 2 122 121 ( ( ( ( ( ( ( : z z z . 0 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0z1z1z1z1z1z1z1z2z3z3z3z3z4z4z4z5 .( A z1I) :1 = 0; ( A z1I) :1g1 = :1; ( A z1I) :1g2 = :1g1; ( A z1I) :2 =0; ( A z1I) :2g1 = :2; ( A z1I) :3 = 0; ( A z1I) :3g1 = :3; ( A z2I) :2 =0; ( A z3I) :3 = 0; ( A z3I) :3g1 = :3; ( A z3I) :3g2 = :3g1; ( A z3I) :3g3 = :3g2

: ) m( z) :( ) ( z) ( A ) A ( : 1 n A ) ( z 1) ( z 2)2 ( z 1)2( z 2) ( ) ( z) = ( z z1)m1( z zn)mn ( m( z) = ( z z1)m1|( z zn)mn| m1i z1 m2i z2 ... m1 m1i z1 ... : ) ( z) = ( z 2)4( z 3)3 m( z) = ( z 2)2( z 3)2 232 1 2 12 22 1 22 23 1 3 13 33 3or (( (( (( (( (( (( (( (( (( (( : 5 5 ( z 2)2 2 :_A100 A2_ ( z) = | A zI| = | A1 zI| | A2 zI| m( z) = ( z z)mi m l l ( z z)Ii | A| z: l: A zI : A zI A r n ( z) = ( z =1z)ni ; n=1= n1 . r = n n AMA; A = Jiog( z1, , zn)2 . r < n; l=1= n n AMA; A = Jiog( z1, , zn) 1 : A 2 : A n A 6 : ) L inearFunctional :( I k . I: I k; I( ox1 + [x2) = oI( x1)+ [I( x2) : ) p( t) I ( . FD: I k . I = { : I k, i = 1, , n} 24 : I n ) : = [ :1, , :n] ( I n In [ : : { }iv I ] I ](:) = o]= _1 ] = i0 ] i : I I : ](:) o] . 1 : R3 :1 = [ 1, 1,3]1; :2 = [ 0,1, 1]1; :3 = [ 0,3, 2]1; . = [ o, b, c] 2 : I 1 = Jt10; 2 = Jt20 I : |:| I ] ) I( u I |:| o I ] :u = ( u) : = 1( u) :1 + 2( u) :2 + ; o = o(:) = o(:1)1 + o(:2)2 + : |:|, |w| I I ] , |o| I I P |:| |w| ] |o| : ( P-1)1 : w ) ( I I w : w w: (w) = 0 : (w) . : w I : I: I u I ) ( I: I k ) ( ) k ( I . I1( ) = I I : I1: I u u: I I; I1: I u; I1( ) = I 25 : : ( x, y) = x 2y R2 I( x, y) = ( x, 0) R2 .I1( ) = I( x, y) = ( x, 0) = x @[1] @ . . x Rn x1 A Rm n m n A = |o]] : A, B Rm n; A + B = |o] + b]] : A + B = B + AA + ( B + C) = ( A + B) + C : : : AB BA : : A( BC) = ( AB) C dydic ) unit dydic (. : r q : rq = rq :Am pBp n = _o11om1 _ [ b1 bn] = _o11b1 o11bn om1 b1 om1 bn_ :ddt(A( t) B( t) ) =ddt(A( t) )B( t) +ddt(B( t) )A( t) : Ax = y 26 : ) ( x x y x + y 0 0 + x = x + 0= x o . o x . x . : x1 xn o1x1 + + onxn = 0 o= 0; i = 1: n . : x x1 xn [ :x = [xn=1 : : n x1 xn y x : y = o1x1 + + onxn o . o y u = |x1, , xn| yu = _o1on_ . ) ( : u1 un w1 wn _ x _ : x = oun=1; x = [wn=1 o [ ) ( :_o1on_ = _p11 p1n pn1 pnn_ _[1[n_ xu = Ixw

:xw = Sxu : IS = I = SI; S = I-1; I = S-1 : . A _ x y u1 un _ x = o1u1 + + onun : y = A(x) =A(o1u1 + + onun) = o1A(u1) + + onA(un) = o1:1 + + on:n : u A : : = y1u1 + + ynun :yu = _y11 y1n yn1 ynn_ _o1on_ yu = Axu

27 : w : yw = Bxw yw = I-1yu : B = I-1AI A B .yu = Axu Iyw = AIxw yw = I-1AIxw B = I-1AI : - A : Ax x . = ]y| x, Ax = y - ) Null Space ( A : x Ax = 0 . N = |x| Ax = 0| 1 : Ax = y y ( A) 2 : Ax = y N( A) = |0| - A : x1A x . = ]y| x, x1A = y - A : x x1A = 0 .K = |x| x1A = 0| rank A : :1 . ( A) = ( A1)

2 . N( A) = K( A1)3 . Amn: Jim(( A) ) = ronk( A) = Jim(( A) )4 . Amn: Jim(N( A) ) = n ronk( A)5 . Amn: Jim(K( A) ) = m ronk( A) : I . : ( A) = K( A)1 N( A) = ( A)1@ ] 3 [