COMPLEX NUMBERS AND QUADRATIC EQUATIONS


 Miles Lamb
 4 years ago
 Views:
Transcription
1 COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s nonnegatve Hence the equatons x, x, x etc are not solvable n real number system Thus, there s a need to extend the real number system to a larger system so that we can have solutons of such equatons In fact, our man objectve s to solve the quadratc equaton ax + bx + c 0, where a, b, c R and the dscrmnant b 4 ac < 0, whch s not possble n real number system In ths chapter, we shall extend the real number system to a larger system called complex number system so that the solutons of quadratc equatons ax + bx + c 0, where a, b, c are real numbers are possble We shall also solve quadratc equatons wth complex coeffcents COMPLEX NUMBERS We know that the equaton x + 0 s not solvable n the real number system e t has no real roots Many mathematcans ndcated the square roots of negatve numbers, but Euler was the frst to ntroduce the symbol (read ota ) to represent, and he defned If follows that s a soluton of the equaton x + 0 Also ( ) Thus the equaton x + 0 has two solutons, x ±, where The number s called an magnary number In general, the square roots of all negatve real numbers are called magnary numbers Thus,, 9 4 etc are all magnary numbers Complex number A number of the form a + b, where a and b are real numbers, s called a complex number For example, +, +, +, 7 + are all complex numbers The system of numbers C { ; a + b ; a, b R} s called the set of complex numbers Standard form of a complex number If a complex number s expressed n the form a + b where a, b R and n the standard form, then t s sad to be For example, the complex numbers +, +, 7 are all n the standard form Real and magnary parts of a complex number If a + b (a, b R) s a complex number, then a s called the real part, denoted by Re () and b s called magnary part, denoted by Im ()
2 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 9 For example : () If +, then Re () and Im () () If +, then Re () and Im () () If 7, then 7 + 0, so that Re () 7 and Im () 0 (v) If, then 0 + ( ), so that Re () 0 and Im () Note that magnary part of a complex number s a real number In a + b (a, b R), f b 0 then a, whch s a real number If a 0 and b 0, then b, whch s called purely magnary number If b 0, then a + b s nonreal complex number Snce every real number a can be wrtten as a + 0, we see that R C e the set of real numbers R s a proper subset of C, the set of complex numbers Note that, 0,, π are real numbers ; +, etc are nonreal complex numbers ;, etc are purely magnary numbers Equalty of two complex numbers Two complex numbers a + b and c + d are called equal, wrtten as, f and only f a c and b d For example, f the complex numbers a + b and + are equal, then a and b Algebra of complex numbers In ths secton, we shall defne the usual mathematcal operatons addton, subtracton, multplcaton, dvson, square, power etc on complex numbers and wll develop the algebra of complex numbers Addton of two complex numbers Let a + b and c + d be any two complex numbers, then ther sum + s defned as + (a + c) + (b + d) For example, let + and + 4, then + ( + ( )) + ( + 4) + 7 Propertes of addton of complex numbers () Closure property The sum of two complex numbers s a complex number e f and are any two complex numbers, then + s always a complex number () Addton of complex numbers s commutatve If and are any two complex numbers, then + + () Addton of complex numbers s assocatve If, and are any three complex numbers, then ( + ) + + ( + ) (v) The exstence of addtve dentty Let x + y, x, y R, be any complex number, then (x + y) + (0 + 0) (x + 0) + (y + 0) x + y and (0 + 0) + (x + y) (0 + x) + (0 + y) x + y (x + y) + (0 + 0) x + y (0 + 0) + (x + y) Therefore, acts as the addtve dentty It s smply wrtten as 0 Thus, for all complex numbers
3 0 MATHEMATICS XI (v) The exstence of addtve nverse For a complex number a + b, ts negatve s defned as ( a) + ( b) a b Note that + ( ) (a a) + (b b) Thus acts as addtve nverse of Subtracton of complex numbers Let a + b and c + d be any two complex numbers, then the subtracton of from s defned as + ( ) (a + b) + ( c d) (a c) + (b d) For example, let + and + 4, then ( + ) ( + 4) ( + ) + ( 4) ( + ) + ( 4) and ( + 4) ( + ) ( + 4) + ( ) ( ) + (4 ) + Multplcaton of two complex numbers Let a + b and c + d be any two complex numbers, then ther product s defned as (ac bd) + (ad + bc) Note that ntutvely, (a + b) (c + d) ac + bc + ad + bd ; now put, thus (a + b) (c + d) ac + (bc + ad) bd (ac bd) + (ad + bc) For example, let + 7 and +, then ( + 7) ( + ) ( ( ) 7 ) + ( + 7 ( )) 4 + Propertes of multplcaton of complex numbers () Closure property The product of two complex numbers s a complex number e f and are any two complex numbers, then s always a complex number () Multplcaton of complex numbers s commutatve If and are any two complex numbers, then () Multplcaton of complex numbers s assocatve If, and are any three complex numbers, then ( ) ( ) (v) The exstence of multplcatve dentty Let x + y, x, y R, be any complex number, then (x + y) ( + 0) (x y0) + ( x0 + y) x + y and ( + 0) (x + y) (x 0y) + (y + 0x) x + y (x + y) ( + 0) x + y ( + 0) (x + y) Therefore, + 0 acts as the multplcatve dentty It s smply wrtten as Thus for all complex numbers (v) Exstence of multplcatve nverse For every nonero complex number a + b, we have the complex number
4 COMPLEX NUMBERS AND QUADRATIC EQUATIONS a a + b b a + b (denoted by or ) such that s called the multplcatve nverse of Note that ntutvely, a b a b a+ b a + b a b a + b a a + b b a + b (check t) (v) Multplcaton of complex numbers s dstrbutve over addton of complex numbers If, and are any three complex numbers, then ( + ) + and ( + ) + These results are known as dstrbutve laws Dvson of complex numbers Dvson of a complex number a + b by c + d 0 s defned as c d ac bd bc ad + ( a + b) + c + d c + d c + d c + d Note that ntutvely, a + b a + b c d ( ac + bd) + ( bc ad) c + d c + d c d c + d For example, f + 4 and 6, then ( ) + ( 6+ 4 ) Integral powers of a complex number If s any complex number, then postve ntegral powers of are defned as,,, 4 and so on If s any nonero complex number, then negatve ntegral powers of are defned as :,, etc If 0, then 0 Powers of Integral power of are defned as : 0,,, ( ), 4 ( ) ( ), 4, 6 4 ( ), and so on
5 MATHEMATICS XI Remember that, 4 4, and so on 4 Note that 4 and 4 It follows that for any nteger k, 4k, 4k+, 4k+, 4k+ Also, we note that and ( ) Therefore, and are both square roots of However, by the symbol, we shall mean only e We observe that and are both the solutons of the equaton x + 0 Smlarly, and ( ) ( ) ( ) ( ) ( ), ( ) Therefore, and are both square roots of However, by the symbol, we shall mean only e In general, f a s any postve real number, then a a We already know that a b ab for all postve real numbers a and b Ths result s also true when ether a 0, b < 0 or a < 0, b 0 But what f a < 0, b < 0? Let us examne : we note that ( )( ) (by assumng a b ab for all real numbers) Thus, we get whch s contrary to the fact that Therefore, a b ab s not true when a and b are both negatve real numbers Further, f any of a and b s ero, then a b ab 0 Identtes If and are any two complex numbers, then the followng results hold : () ( + ) + + () ( ) + () ( + ) ( ) (v) ( + ) (v) ( ) + Proof () ( + ) ( + ) ( + ) ( + ) + ( + ) (Dstrbutve law) (Dstrbutve law) (Commutatve law) + + We leave the proofs of the other results for the reader
6 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 4 Modulus of a complex number Modulus of a complex number a + b, denoted by mod() or, s defned as a + b, where a Re (), b Im () Sometmes, s called absolute value of Note that 0 For example : () If +, then ( ) + 4 () If 7, then + ( 7 ) Propertes of modulus of a complex number If, and are complex numbers, then () () 0 f and only f 0 () (v), provded 0 Proof () Let a + b, where a, b R, then a b ( a) + ( b) a + b () Let a + b, then a + b () Let Now 0 ff a + b 0 e ff a + b 0 e ff a 0 and b 0 e ff a 0 and b 0 e ff e ff 0 a + b, and c + d, then (ac bd) + (ad + bc) ( ac bd) + ( ad + bc) a c + b d abcd + a d + b c + abcd ( a + b )( c + d ) a + b c + d ( Q a + b 0, c + d 0) (v) Here 0 0 Let (usng part ()) Q REMARK From (), on replacng both and by, we get Smlary, e etc
7 4 MATHEMATICS XI Conjugate of a complex number Conjugate of a complex number a + b, denoted by, s defned as a b e a+ b a b For example : () +, + () 7 + 7, Propertes of conjugate of a complex number If, and are complex numbers, then () ( ) () + + () (v) (v), provded 0 (v) (v) (v) Proof () Let a + b, where a, b R, so that a b ( ) a b a + b () Let a + b and c + d, then + ( a + b) + ( c + d) ( a + c) + ( b + d) (a + c) (b + d) (a b) + (c d) + () Let a + b and c + d, then ( a+ b) ( c + d) ( a c) + ( b d) (a c) (b d) (a b) (c d), provded 0 (v) Let a + b and c + d, then ( a + b)( c + d) ( ac bd) + ( ad + bc) (ac bd) (ad + bc) Also (a b) (c d) (ac bd) (ad + bc) Hence (v) Here 0 0 Let (usng part (v)) Q (v) Let a + b, then a b a + ( b) a + b
8 COMPLEX NUMBERS AND QUADRATIC EQUATIONS (v) Let a + b, then a b (a + b) (a b) (aa b( b)) + (a( b) + ba) (Def of multplcaton) (a + b ) + 0 a + b ( a + b ) Remember that (a + b) (a b) a + b (v) Let a + b 0, then 0 (a + b) (a b) a + b Thus,, provded 0 REMARK From (v), on replacng both and by, we get e ( ) Smlarly, ( ) ( ) ( ) ( ) ( ) etc NOTE The order relatons greater than and less than are not defned for complex numbers e the nequaltes + +, 4, + < etc are meanngless ILLUSTRATIVE EXAMPLES Example A student says ( ) ( ) Thus Where s the fault? Soluton ( ) ( ) s true, but ( )( ) s wrong Because f both a, b are negatve real numbers, then a b ab s not true Example If 7 + 9, fnd Re(), Im (), and Soluton Gven Re () 7 and Im() ( 7) + ( 9) Example If 4x + (x y) 6 and x, y are real numbers, then fnd the values of x and y Soluton Gven 4x + (x y) 6 4x + (x y) + ( 6)
9 6 MATHEMATICS XI Equatng real and magnary parts on both sdes, we get 4x and x y 6 x 4 and y 6 4 x 4 and y Hence x 4 and y 4 Example 4 For what real values of x and y are the followng numbers equal () ( + ) y + (6 + ) and ( + ) x () x 7 x + 9 y and y + 0? Soluton () Gven ( + ) y + (6 + ) ( + ) x ( y + 6) + (y + ) x + x y + 6 x and y + x x and y 4 x and y ± Hence, the requred values of x and y are x, y ; x, y () Gven x 7 x + 9 y y + 0 (x 7 x) + (9 y) ( ) + ( y + 0) x 7 x and 9 y y + 0 x 7 x + 0 and y 9 y (x 4) (x ) 0 and (y ) (y 4) 0 x 4, and y, 4 Hence, the requred values of x and y are x 4, y ; x 4, y 4; x, y ; x, y 4 Example Express each of the followng n the standard form a + b : 7 4 () () (7 + 7) + (7 + 7) () ( + ) ( + Soluton () ) (v) 7 + () (7 + 7) + (7 + 7) ( + ) + (7 + 7 ) ( ) ( 7) + ( + 7) () ( + )( + ) ( + ) ( + ) (6 ) + ( 4 ) 0 7 ( + ) ( ) ( + ) ( )
10 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 7 (v) ( + ) ( ) ( + ) ( ) ( ) ( ) + + ((a + b) (a b) a + b ) ( ) Q 0 7 Example 6 Express the followng n the form a + b : () ( ) () 8 () 0 () 9 (v) ( ) (v) (v) Soluton () ( ) () 8 ( ) () ( Q 4k +, k I ) + 0 () 9 4 ( 0) + ( Q 4k +, k I) 0 + (v) ( ) ( ) ( ) ( Q 4k +, k I) ( ) 0 + (v) ( ) (v) ( 9) ( ) Example 7 Express each of the followng n the standard form a + b : () ( ) 4 () () ( ) + ( ) (v) (v) ( + ) 6 + ( ) Soluton () ( ) 4 (( ) ) ( + ) ( + ( ) ) ( ) 4 4 ( ) (NCERT Examplar Problems)
11 8 MATHEMATICS XI () ( ) ( ) ( ) () ( ) + ( ) ( + ) + ( ) (v) 8 + ( ) + ( ) ( ) ( ) Q ( ) + ( ) ( ) ( ) ( ) ( + ) [ ] [ ] + ( + ) (v) ( + ) 6 (( + ) ) ( + + ) ( + ) () 8 8 ( ) 8 and ( ) + ( ) + ( ) ( + ) 6 + ( ) 8 + ( ) 0 Example 8 Fnd the multplcatve nverse of + Soluton Let +, then and ( ) We know that the multplcatve nverse of s gven by Alternatvely + + ( ) ( ) 9 ( ) Example 9 Express the followng n the form a + b : () () + + () 4 4 (v) + + ( + )( + ) ( )( ) +
12 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 9 Soluton () + + ( ) ( ) + + () ( ) + 6 ( ) () ( 4) ( + ) ( + 9 ) ( 4 ) ( ) ( + ) ( 8) ( 0) ( ) ( ) (v) ( + )( + ) ( )( ) ( + )( + )( + ) ( )( )( ) + ( )( + ) ( + )( ) ( )( 8 6 ) 9 ( + 4 ) ( 4 ) Example 0 () If ( + ) x + y, then fnd the value of x + y (NCERT Examplar Problems) () If + x + y, then fnd (x, y) (NCERT Examplar Problems) + () If + 00 a + b, then fnd (a, b) (NCERT Examplar Problems) Soluton () x + y ( + ) ( ) + 4 ( ) x and y 4 4 x + y + 4 () We have, ( + ) ( ) + Q
13 40 MATHEMATICS XI Gven x + y + + ( ) + ( ) 0 x 0 and y Hence, the orders par (x, y) s (0, ) () Gven a + b + 00 ( ) 00 () a and b 0 Hence, the ordered par (a, b) s (, 0) ( ) 00 (see part ()) Example () If ( + ) ( ), then show that (NCERT Examplar Problems) () If and +, fnd Re Soluton () Gven ( + ) ( ) ( ) ( ) () Re ( )( ) ( ) ( ) 4 ( ) + + Example () Fnd the conjugate of ( ) ( + ) ( + ) ( ) () Fnd the modulus of + + () Fnd the modulus of ( ) + Soluton () Let ( ) ( + ) ( + ) ( ) ( ) ( ) Conjugate of 6 6 +
14 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 49 Example If α and β are dfferent complex numbers wth β, then fnd β α αβ Soluton We have β α ( β α) β (Note ths step) αβ ( αβ) β ( β α ) β β αββ ( β α ) β β α (Gven β β ββ ) Hence, ( β α ) β β α Q β α β β α ( Q ) β α β β α ( Q ) β ( β, gven) β α αβ Example If n, prove that n (NCERT Examplar Problems) n Soluton Gven n n,,, n n,,, n n n n ( Q ) n n EXERCISE Very short answer type questons ( to ) : Evaluate the followng : () 9 4 () ( 9) ( 4 ) () 6 (v) 6 (v) If, fnd Re (), Im(), and If, then s t true that ± ( )? 4 If + 4, then s t true that ± ( + )? If, then show that (x + + ) (x + ) x + x + 6 Fnd real values of x and y f () y + (x y) () (x ) + ( + y) () (y ) + (7 x) 0
15 0 MATHEMATICS XI 7 If x, y R and ( y ) + ( x y) 7, fnd the values of x and y 8 If x, y are reals and ( y + ) + (x + y) 0, fnd the values of x and y 9 If x, y are reals and ( ) x + ( + ) y, fnd the values of x and y 0 For any two complex numbers and, prove that Re ( ) Re ( ) Re( ) Im ( ) Im ( ) For any complex number, prove that Re() + and Im() If s a complex number, show that s real Express the followng ( to 0) complex numbers n the standard form a + b : () ( ) 8 () 4 () ( ) ( + 6 ) () ( ) + () ( + ) + + ( ) () (7 + ) (7 ) 6 () ( + ) ( ) () ( + ) 7 () () ( ) 8 () + () ( + 7) ( 7) 9 () 99 () 0 () ( 4 ) () Fnd the value of ( + ) + ( ) If n s any nteger, then fnd the value of () ( ) 4n + () 4n+ 4n (NCERT Examplar Problems) Fnd the multplcatve nverse of 4 Express the followng numbers n the form a + b, a, b R : () () + + If (a + b) (c + d) A + B, then show that (a + b ) (c + d ) A + B 6 Fnd the modulus of the followng complex numbers : () ( 4 ) ( + ) () Fnd the modulus of the followng : () ( ) 4 + () ( 7 ) 8 () If 7, then fnd () If x + y, x, y R, then fnd 9 Fnd the conjugate of 7 0 Wrte the conjugate of ( + ) ( ) n the form a + b, a, b R Solve for x : + x Express the followng ( and ) complex numbers n the standard form a + b : () ()
16 MATHEMATICS XI 44 If a + b ( x + ) x +, then prove that a + b ( x + ) ( x + ) 4 If +, then fnd the value of Show that f, then s a real number + 47 () If x + y and, show that (x + y ) 0x + 0 (NCERT Examplar Problems) () If x + y and + 4, show that x + y x y 7 n + 48 Fnd the least postve ntegral value of n for whch s a real number 49 Fnd the real value of θ such that + cos θ s a real number cos θ 0 If s a complex number such that, prove that ( ) s purely magnary + What s the excepton? (NCERT Examplar Problems) If, and are complex numbers such that + +, then fnd the value of + + (NCERT Examplar Problems) ARGAND PLANE We know that correspondng to every real number there exsts a unque pont on the number lne (called real axs) and conversely correspondng to every pont on the lne there exsts a unque real number e there s a oneone correspondence between the set R of real numbers and the ponts on the real axs In a smlar way, correspondng to every ordered par (x, y) of real numbers there exsts a unque pont P n the coordnate plane wth x as abscssa and y as ordnate of the pont P and conversely correspondng to every pont P n the plane there exsts a unque ordered par of real numbers Thus, there s a oneone correspondence between the set of ordered pars {(x, y) ; x, y R} and the ponts n the coordnate plane The pont P wth coordnates (x, y) s sad to represent the complex number x + y It follows that the complex number x + y can be unquely represented by the pont P(x, y) n the coordnate plane and conversely correspondng to the pont P(x, y) n the plane there exsts a unque complex number x + y The coordnate plane that represents the complex numbers s called the complex plane or Argand plane The complex numbers,,,, +,, + and whch correspond to the ordered pars (, 0), (, 0), (0, ), (0, ), (, ), (, ), (, ) and (, ) respectvely have been represented geometrcally n the coordnate plane by the ponts A, B, C, D, E, F, G and H respectvely n fg X X O < + O B (, 0) x Fg G (, ) < C (0, ) y D (0, ) H (, ) Fg x + y P(x, y) + E (, ) A X (, 0) F (, ) X
17 COMPLEX NUMBERS AND QUADRATIC EQUATIONS Note that every real number x x + 0 s represented by pont (x, 0) lyng on xaxs, and every purely magnary number y s represented by pont (0, y) lyng on yaxs Consequently, xaxs s called the real axs and yaxs s called the magnary axs If the pont P(x, y) represents the complex number x + y, then the dstance between the ponts P and the orgn O (0, 0) x + y Thus, the modulus of e s the dstance between ponts P and O (see fg ) X O < x + y P(x, y) < y x Fg X Geometrc representaton of, and If x + y, x, y R s represented by the pont P(x, y) n the complex plane, then the complex numbers,, are represented by the ponts P ( x, y), Q(x, y) and Q ( x, y) respectvely n the complex plane (see fg 4) X Q ( x, y) O P(x, y) X Geometrcally, the pont Q (x, y) s the mrror mage of the pont P (x, y) n the real axs Thus, conjugate of e s the mrror mage of n the xaxs P ( x, y) Fg 4 Q(x, y) POLAR REPRESENTATION OF COMPLEX NUMBERS Let the pont P (x, y) represent the nonero complex number x + y n the Argand plane Let the drected lne segment OP be of length r ( 0) and θ be the radan measure of the angle whch OP makes wth the postve drecton of xaxs (shown n fg ) Then x + y P(x, y) r x + y and s called modulus of ; and θ s called ampltude or argument of and s wrtten as amp() or arg() From fgure, we see that x r cos θ and y r sn θ x + y r cos θ + r sn θ r (cos θ + sn θ) r θ O < x M Fg Thus, r (cos θ + sn θ ) Ths form of s called polar form of the complex number X < y < X P P r r X O < θ X X < O θ X () ()
18 4 MATHEMATICS XI X r < O θ X X O < θ r X P P () (v) Fg 6 For any nonero complex number, there corresponds only one value of θ n π < θ π (see fg 6) The unque value of θ such that π < θ π s called prncpal value of ampltude or argument Thus every (nonero) complex number x + y can be unquely expressed as r (cos θ + sn θ ) where r 0 and π < θ π and conversely, for every r 0 and θ such that π < θ π, we get a unque (nonero) complex number r (cos θ + sn θ ) x + y Note that the complex number ero cannot be put nto the form r (cos θ + sn θ) and so, the argument of ero complex number does not exst REMARK If we take orgn as the pole and the postve drecton of the xaxs as the ntal lne, then the pont P s unquely determned by the ordered par of real numbers (r, θ), called the polar coordnates of the pont P (see fg 6) ILLUSTRATIVE EXAMPLES Example Convert the followng complex numbers n the polar form and represent them n Argand plane : () + () + () (v) (v) (v) Soluton () Let + r (cos θ + sn θ) Then r cos θ and r sn θ On squarng and addng, we get r (cos θ + sn θ) ( ) + r 4 r cos θ and sn θ The value of θ such that π < θ π and satsfyng both the above equatons s gven by θ π 6 X O < π/6 + P X π π Hence, cos + sn, whch s the requred polar 6 6 form The complex number + s represented n fg 7 Fg 7
19 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 7 ANSWERS EXERCISE () 6 () 6 () 0 (v) 60 (v) 0 ; ; + ; 0 es 4 es 6 () x 6, y () x, y () x 7, y 7 x, y 8 x, y 9 x, y () () () 7 () 0 () () () ( 6 + ) + ( + ) () () 8 () () () () 0 () () () () () 4 () + () 0 6 () 6 () 7 () () 64 8 () 4 () x + y () () 0 + () () () () (v) () () 4 47 () 4 + (v) 6 9 ; () 6 4 () + (v) ; () + ; ; () + ; ; (v) ; 9 46 ; 97 7 () 0 + () (v) n 8 () 9 () x, y 4 () () x 6, y () x () 0, y 8 40 () s the only soluton () all purely magnary numbers (n + ) π, n I 0 Excepton s EXERCISE () True () True () True (v) True (v) True 0 θ 4 () + () () 0 (v) + +
332600_08_1.qxp 4/17/08 11:29 AM Page 481
336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationComplex Numbers. x = B B 2 4AC 2A. or x = x = 2 ± 4 4 (1) (5) 2 (1)
Complex Numbers If you have not yet encountered complex numbers, you wll soon do so n the process of solvng quadratc equatons. The general quadratc equaton Ax + Bx + C 0 has solutons x B + B 4AC A For
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationLectures  Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures  Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCCThe Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 50757 HIKARI Ltd, wwwmhkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationSTUDY PACKAGE. Subject : Mathematics Topic : COMPLEX NUMBER Available Online :
fo/u fopkjr Hkh# tu] ugha vkjehks dke] fofr ns[k NksM+s rqjar e/;e eu dj ';kea q#"k flag lady dj] lgrs fofr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez.ksrk ln~xq# Jh j.knksm+nklth
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3d rotaton around the xaxs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3d rotaton around the yaxs:
More informationof Nebraska  Lincoln
Unversty of Nebraska  Lncoln DgtalCommons@Unversty of Nebraska  Lncoln MAT Exam Expostory Papers Math n the Mddle Insttute Partnershp 008 The Square Root of Tffany Lothrop Unversty of NebraskaLncoln
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationImportant Instructions to the Examiners:
Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as wordtoword as gven n the model
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented
More informationComplex Numbers Alpha, Round 1 Test #123
Complex Numbers Alpha, Round Test #3. Wrte your 6dgt ID# n the I.D. NUMBER grd, leftjustfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3dgt Test
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationOn cyclic of Steiner system (v); V=2,3,5,7,11,13
On cyclc of Stener system (v); V=,3,5,7,,3 Prof. Dr. Adl M. Ahmed Rana A. Ibraham Abstract: A stener system can be defned by the trple S(t,k,v), where every block B, (=,,,b) contans exactly Kelementes
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the righthand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The nonrelativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The nonrelatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)
7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to
More information(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1)
. (a) (cos θ + sn θ) = cos θ + cos θ( sn θ) + cos θ(sn θ) + (sn θ) = cos θ cos θ sn θ + ( cos θ sn θ sn θ) (b) from De Movre s theorem (cos θ + sn θ) = cos θ + sn θ cos θ + sn θ = (cos θ cos θ sn θ) +
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationIndeterminate pinjointed frames (trusses)
Indetermnate pnjonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationGroup Theory Worksheet
Jonathan Loss Group Theory Worsheet Goals: To ntroduce the student to the bascs of group theory. To provde a hstorcal framewor n whch to learn. To understand the usefulness of Cayley tables. To specfcally
More informationComplex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: ShihYuan Chen
omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: ShhYuan hen Except where otherwse noted, content s lcensed under a BYNSA. TW Lcense. ontents ontour ntegrals auchygoursat theorem
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac KTheory
More informationMath 261 Exercise sheet 2
Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 BCH Code Consder the [15, 7, 5] 2 code C we ntroduced n the last lecture
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
.1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fntedmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscretetme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationThe Second AntiMathima on Game Theory
The Second AntMathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2player 2acton zerosum games 2. 2player
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 7145744776
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space
More information3.1 Expectation of Functions of Several Random Variables. )' be a kdimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationSome basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C
Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +
More informationSome Inequalities Between Two Polygons Inscribed One in the Other
BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. () 8() (005), 73 79 Some Inequaltes Between Two Polygons Inscrbed One n the Other Aurelo de Gennaro
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 4: Fundamental Theorem of Algebra. Instruction. Guided Practice Example 1
Guded Practce 3.4. Example 1 Instructon For each equaton, state the number and type of solutons by frst fndng the dscrmnant. x + 3x =.4x x = 3x = x 9x + 1 = 6x 1. Fnd the dscrmnant of x + 3x =. The equaton
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationFor all questions, answer choice E) NOTA" means none of the above answers is correct.
0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationU.C. Berkeley CS294: Beyond WorstCase Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond WorstCase Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationThe Jacobsthal and JacobsthalLucas Numbers via Square Roots of Matrices
Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513520 HIKARI Ltd, wwwmhkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and JacobsthalLucas Numbers va Square Roots of Matrces Saadet Arslan
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v1006500800206 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More informationhanalogue of Fibonacci Numbers
hanalogue of Fbonacc Numbers arxv:090.0038v [mathph 30 Sep 009 H.B. Benaoum Prnce Mohammad Unversty, AlKhobar 395, Saud Araba Abstract In ths paper, we ntroduce the hanalogue of Fbonacc numbers for
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr GyeSeon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationExercises. 18 Algorithms
18 Algorthms Exercses 0.1. In each of the followng stuatons, ndcate whether f = O(g), or f = Ω(g), or both (n whch case f = Θ(g)). f(n) g(n) (a) n 100 n 200 (b) n 1/2 n 2/3 (c) 100n + log n n + (log n)
More information1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52
ontents ommonly Used Formulas. Estmaton, pproxmaton and Errors. Percentages. Polynomals and Formulas 8. Identtes and Factorzaton. Equatons and Inequaltes 66 6. Rate and Rato 8 7. Laws of Integral Indces
More informationCHAPTER5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationAGC Introduction
. Introducton AGC 3 The prmary controller response to a load/generaton mbalance results n generaton adjustment so as to mantan load/generaton balance. However, due to droop, t also results n a nonzero
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions
Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a twodgt number, we have
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationSome Consequences. Example of Extended Euclidean Algorithm. The Fundamental Theorem of Arithmetic, II. Characterizing the GCD and LCM
Example of Extended Eucldean Algorthm Recall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to wrte 3 as a lnear combnaton of 84 and 33: 3 = 18 15 [Now 3 s
More information3 Basic boundary value problems for analytic function in the upper half plane
3 Basc boundary value problems for analytc functon n the upper half plane 3. Posson representaton formulas for the half plane Let f be an analytc functon of z throughout the half plane Imz > 0, contnuous
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
More informationSpinrotation coupling of the angularly accelerated rigid body
Spnrotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 Emal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationThe Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions
56 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9,
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationMathematics Intersection of Lines
a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warmup Show that the egenvalues of a Hermtan operator Â are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationNPCompleteness : Proofs
NPCompleteness : Proofs Proof Methods A method to show a decson problem Π NPcomplete s as follows. (1) Show Π NP. (2) Choose an NPcomplete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationQuantum Mechanics I  Session 4
Quantum Mechancs I  Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. > Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationTHERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q.
THERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q. IAN KIMING We shall prove the followng result from [2]: Theorem 1. (BllngMahler, 1940, cf. [2]) An ellptc curve defned over Q does not have a
More information