Lecture 2: Matrix Algebra. General

Size: px
Start display at page:

Download "Lecture 2: Matrix Algebra. General"

Transcription

1 Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots nd Vectors Trce Qudrtic Forms Generl A mtrix is rectngulr rry of objects or elements. We will tke these elements s being rel numbers nd indicte n element by its row nd column position. Let ij R denote n element of mtrix which occupies the position of the ith row nd jth column. Denote mtrix by cpitl letter nd its elements by the corresponding lower cse letter. If mtrix A is nx m, we write A nxm 1

2 Generl Exmple 1. A 2 2 Exmple 2. A Exmple 3. A n n n 11 1n 1 nn Exmple 4. A Generl A mtrix is sid to be (i) squre if # rows = # columns nd squre mtrix is sid to be (ii) symmetric if ij = ji i, j, i j. Exmple. The mtrix is squre but not symmetric, since 21 = 2 3 = 12. The squre m trix is sym m etric since 12 = 21 = 2, 31 = 13 = 4, nd 32 = 23 =

3 Generl The principle digonl elements of squre mtrix ti A re given by the elements ij, i = j. The principle digonl is the ordered n- tuple ( 11,..., nn ). The trce of squre mtrix is defined s the sum of the principl digonl elements. It is denoted tr(a) = i ii. Exmple principl digonl is (1,1,1), Tr(A) = 3 A =

4 Generl A digonl mtrix is squre mtrix whose only nonzero elements pper on the principl digonl. A sclr mtrix is digonl mtrix with the sme vlue in ll of the digonl elements. Exmples: Digonl : A Sclr : A Generl The identity mtrix is sclr mtrix with ones on the digonl. A tringulr mtrix is squre mtrix tht hs only zeros either bove or below the principl digonl. If the zeros re bove the digonl, then the mtrix is lower tringulr nd conversely for upper tringulr. 4

5 5 Exmples upper tringulr lower tringulr Nottion The following nottions for indicting n n m mtrix A re equivlent The following nottions for indicting n n m mtrix A re equivlent or ij i n j m n1 nm m,...,,...,,,,.

6 Generl If mtrix A is of dimension 1 n, then it is termed row vector, A 1 n 11 1n 1 n only one row, the row index is sometimes dropped nd A is written.. Since there is 11 A mtrix A of dimension n 1 is termed column vector, A. Likewise, since there is n 1 1 only one column, this is sometimes written s. n Algebric Opertions on Mtrices Equlity: A = B if ij = b ij for ll i nd j. Addition nd Subtrction: A ± B = [ ij ±b ij ]. Note tht for these opertions, A nd B must be of the sme dimension. A nm b b b b m 1m B AB n1 bn1 nm bnm 1b 1 b n m m 1m n n nm nm. 6

7 Algebric Opertions on Mtrices Sclr multipliction:. Let k R. ka = [k ij ] Algebric Opertions on Mtrices: Multipliction Conformbility: Two mtrices A nd B cn be multiplied li to form AB, only if the column dimension of A = row dimension of B. (col. dim. led = row dim. of lg) Exmple If A 23 nd B, then AB cnnot be defined, but BA cn be defined

8 Multipliction Inner product of two n-tuples: Suppose x, y R n. Then the inner product (lso clled the dot product) of x nd y is defined by n i i i 1 x y x y x y x y x y n n Multipliction Associte with the kth col of A (n x m) the n-tuple ok = ( 1k,, nk ) R n Associte with the jth row of A the m-tuple jo = ( j1,, jm ) R m Exmple. A = (2, 4) 20 = (0, 4, 5) 8

9 Multipliction The product AB is then given by A B nm mk b b b b k n0 01 n0 0k nxk m i1 m b 1i i1 b m i1 m ni i1 i1 i 1 b 1i b ni ik ik Multipliction Note tht the product mtrix is n x k. It tkes on the row dimension i of the led nd the column dimension of the lg. Exmple: A 2 1 B AB x

10 Multipliction: Exmple A B b b b b b b b b b b b b b b 1i i1 i 1 i i1 2 1i i2 b b 2i i1 i1 2 i i2 b b b b Sclr product: Multipliction Suppose tht A is l n row vector A = = ( n ) nd B n n 1 col vector b11 Bb. Hence we hve b n 1 b11 b 11 1n 1nn1 bn 1 b 1i i1 i 11 10

11 Multipliction: Sclr Product Continued Note tht b = b where,b R n (The sclr product is sme s the inner product of two equivlent ordered n- tuples.) Let i be column vector of ones nd x n n x 1 column vector, then i'x = i x i. Multipliction: Specil Cse The product of conformble column vector (m x 1) nd row vector (1 x n) is n m x n mtrix: b b 11 b b 11 1n 1 n b m1 b m1 b m1 m n 11 1n m n. 11

12 Addition nd Multipliction: Properties The opertion of ddition is both commuttive nd ssocitive. We hve (Com. Lw) A + B = B + A (Associtive) (A + B) + C = A + (B + C) The opertion of multipliction is not commuttive but it does stisfy the ssocitive nd distributive lws. (Associtive) (AB)C = A(BC) (Distributive) A(B + C) = AB + AC (B + C) A = BA + CA Multipliction is not commuttive To see tht AB BA consider the exmple A 1 2 B 0 1, We hve tht AB BA

13 Eqution Systems Generlly, when we tke the product of mtrix ti nd vector, we cn write the result s c = Ab. In this exmple, the mtrix A is n by n nd the column vectors c nd b re n by 1. Eqution Systems Tking n exmple of 22 mtrix A, we hve b This cn be short-hnd wy to write two equtions in the unknowns nd b 1 = + 3b 4 = 3 + 2b. 13

14 Eqution Systems This sme system cn be written s liner combintion of the columns of A b Trnspose The trnspose of mtrix A, denoted A, is the mtrix formed by interchnging i the rows nd columns of the originl mtrix A. Exmple 1. Let A = (1 2) then A Exmple 2. Let A 3 4, then A

15 Key Properties of Trnspose 1. (A) = A 2. (A + B) = A + B 3. (AB) = BA The Identity Mtrix An identity mtrix is squre mtrix with ones in its principle digonl nd zeros elsewhere. An n n identity mtrix is denoted I n. For exmple I

16 Properties of I n 1. Let A be n p. Then we hve I n A = AI p = A. 2. Let A be n p nd B be p m. Then we hve 3. A I p B A I B A B. n p p p p m I I I I I. n n n n n p t erm s In generl, mtrix is termed idempotent, when it stisfies the property AA = A. p The Null Mtrix The null mtrix, denoted [0] is mtrix whose elements re ll zero. 16

17 The Null Mtrix: Properties 1. A + [0] = [0] + A = A 2. [0]A = A[0] = [0]. 3. Remrk: If AB = [0], it need not be true tht A = [0] or B [0]. Exmple where AB = 0: A B Determinnts nd Relted Concepts. A determinnt is defined only for squre mtrices. ti When tking the determinnt tof mtrix we ttch sign + or - to ech element: sign ttched dto i+j ij = sign (-1). 17

18 Determinnts The determinnt of sclr x, is the mtrix itself. The determinnt of 2 2 mtrix A, denoted A or det A, is defined s follows: A Determinnts Exmple 3 6 A A

19 Determinnts n x n: Lplce Expnsion process Definition. The minor of the element ij, denoted d M ij i is the determinnt tof fthe submtrix formed by deleting the ith row nd jth column. Exmple: If A = [ ij ]is3x3 3, then M 13 = M 12 = Determinnts n x n: Lplce Expnsion process Definition. The cofctor of the element ij denoted d C ij given by (-1) i+j M ij. Exmple: In the bove 3 x 3 exmple C 13 = C 12 =

20 Determinnts n x n: Lplce Expnsion process Lplce Expnsion: Let A be n n, n 2. Then A n i1 ij C ij (expnsion by j th col) A C n j1 ij ij (expnsion by i th row) Exmples A is Wht is A? (nswer: -3) 20

21 Properties of Determinnts 1. A = A' 2. The interchnge of ny two rows (or two col.) will chnge the sign of the determinnt, but will not chnge its bsolute vlue. Exmples of Properties 1 nd 2 #1 A A A 2 4 A2 #2 A 1 2, B A 2 B 2 21

22 Properties of Determinnts 3. The multipliction of ny p rows (or col) of mtrix A by sclr k will chnge the vlue of the determinnt to k p A. 4. The ddition (subtrction) of ny multiple of ny row to (from) nother row will leve the vlue of the determinnt unltered, if the liner combintion is plced in the initil (the trnsformed) row slot. The sme holds true if we replce the word row by column. Exmples of Properties 3 nd 4 Tke A, 2 x 2, nd multiply by 2. 2A = = 4 A Tke A, 2 x 2, nd dd 2 times the second row to the first row. ~ A 11 2 ~ , A

23 Properties of Determinnts 5. If one row (col) is multiple of nother row (col), the vlue of the determinnt t will be zero. 6. If A nd B re squre, then AB = A B. Exmples of Properties 5 nd 6 Let Let 3 3 3b A, A 3b 3b 0 b A A 3, B B,, AB, AB , A B 6. 23

24 Def. Define Vector Spces, Liner Independence nd Rnk An n-component vector is n ordered n tuple of rel numbers written s row 1 n or s col the vector. 1. The, i 1,, i n, n re termed the compo nents o f The elements of such vector cn be viewed s the coordintes of point in R n or s the definition of the line segment connecting the origin nd this point. We will tke these s ordered n-tuples: ( 1,, n ) R n Two bsic opertions Sclr multipliction: k = (k 1,,k n ) k 2 (k > 0) (k < 0)

25 Two bsic opertions Addition: + b = ( 1 + b 1,, n + b n ) 2 + b = c b 1 Vector Spce Def. A vector spce is collection of vectors tht t is closed under the opertions of ddition nd sclr multipliction. Remrk: R n is vector spce. Def. A set of vectors spn vector spce if ny vector in tht spce cn be written s liner combintion of the vectors in tht set. 25

26 Bsis A set of vectors spnning vector spce which h contins the smllest number of vectors is clled bsis. This set must be must be linerly independent. If the set were dependent, then some one could be expressed s liner combintion of the others nd it could be eliminted. In this cse we would not hve the smllest set. Liner Independence Def. A set of vectors 1,, m is sid to be linerly l dependentd if i R not ll zero such tht m m = (0,,0) R n. If the only set of i for which this holds is i = 0, for ll i, then the set 1,, m is sid to be linerly independent. 26

27 Results Proposition 1. The vectors 1,, n from R n re linerly dependent iff some one of the vectors is liner combintion of the others. Proof: (i) Let 1 be lin combo of the others nd show dependence. (ii) Assume dependence d nd show tht k cn be written s lin combo of the others. Specil Cse Remrk: If the set of vectors hs but one member R n, then is linerly l dependent if = 0 nd is linerly independent, if 0. 27

28 Results Proposition 2. No set of linerly independent d vectors cn contin the zero vector. Proof: Let 1 = 0 n. Set 1 = 1 nd ll others = 0. Results Proposition 3. Any subset of set of linerly l independent d vectors is linerly l independent. Proof: Assume to the contrry tht 1,,k of i re linerly dependent nd show tht they ll re. 28

29 Results Proposition 4. Any superset of set of linerly l dependent d vectors is linerly l dependent. Proof: Use direct proof. Def of Bsis Def. A bsis for vector spce of n dimensions i is ny set of n linerly l independent vectors in tht spce. In R n, exctly n vectors cn form bsis. Tht is, it tkes n (independent) vectors to crete ny other vector in R n through liner combintion. 29

30 Exmple Let,b be linerly independent in R 2. Let c be ny third vector. We cn show tht t c cn be creted from liner combintion of nd b. Select 1 nd 2 such tht b = c. Tht is i b i = c i,i= 1,2. We hve 1 = (b 2 c 1 b 1 c 2 )/(b 2 1 b 1 2 ) nd 2 = ( 1 c 2 c 1 2 )/ ( 1 b 2 b 1 2 ). Exmple To solve for i, it must be true tht ( 1 b 2 b 1 2 ) 0. This is true if 1 / 2 b 1 /b 2. Tht is, cn not be scle multiple of b. 30

31 Rnk Def. The rnk of n n m mtrix A, r(a), is defined d s the lrgest # of linerly l independent columns or rows. Proposition 1. Given n n m mtrix A, we hve (i) r(a) min {m, n} (ii) lrgest # lin indep. col. = lrgest # lin indep. rows. An Opertionl Test for Rnk Proposition 2. The rnk, r(a), of n m n mtrix A is equl to the order of the lrgest submtrix of A whose determinnt is nonzero. (By submtrix we men mtrix selected from A by eliminting rows nd columns of A.) An n x n mtrix with nonvnishing det hs n linerly independent rows or columns. 31

32 Exmple Determine tht the rnk is Inverse Mtrix Def. Given n n n squre mtrix A, the inverse mtrix of fa, denoted da -1, is tht t mtrix which stisfies A -1 A = A A -1 = I n. When such mtrix exists, A is sid to be nonsingulr. If A -1 exists it is unique. 32

33 Result Proposition. An n x n mtrix A is nonsingulr iff r(a) = n. Computtion of Inverse Assume tht A is n x n nd hs A 0. Cofctor mtrix of A is C =[ C ij ]. The djoint mtrix is dj A = C'. A -1 = (dj A) / A. 33

34 Exmple Compute the inverse of A / 25 3/ 25 A / 25 1/ 25 Key Properties (AB) -1 = B -1 A -1 Proof: B -1 A -1 AB = I nd ABB -1 A -1 = I. (A -1 ) -1 = A Proof: AA -1 = I nd A -1 A = I. I -1 = I Proof: II = I 34

35 Remrks Note tht AB = 0 does not imply tht A = 0 or tht B = 0. If either A or B is nonsingulr nd AB = 0, then the other mtrix is the null mtrix. AB = 0 nd A 0 B = 0 Proof : Let A 0 nd AB = 0. Then A -1 AB = B = 0. Remrks If A nd B re squre, then AB = 0 iff A = 0, B = 0 or both. Proof: Note tht AB = A B. 35

36 Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule. Let A be n x n, let x be column vector of unknown vribles nd let d be column vector of constnts. Ax = d is liner eqution system of n equtions in n unknowns, x i. A unique solution is possible if A 0in which cse A -1 exists. Tht is the rows nd columns of A re linerly independent. Solution The solution cn be written A -1 Ax = A -1 d x = A -1 d Alterntively, define A j = j th col 11 n 1 d d 1 n 1 n nn 36

37 Solution We hve x j = A j / A. (Crmer's Rule) Solve Exmple 3x 1 + 4x 2 = 10 6x 1 + x 2 = 20 Answer: x 1 =70/21 nd x 2 = 0 37

38 Chrcteristic Roots Let D be n n n mtrix. Does there exist sclr r nd n n 1 vector x 0 such tht Dx = rx? If so, then r is sid to be chrcteristic root of D. Rewrite Chrcteristic Roots Chrcteristic Mtrix of D (*) [D - ri]x = 0. For (*) to be true it is necessry tht D - ri = 0, given tht x 0. Proof: To see tht this is true, let A = [D - ri] nd suppose to the contrry tht Ax = 0, x 0, nd A 0. Then A -1 Ax = 0 nd x = 0, so tht we hve contrdiction. 38

39 The condition Chrcteristic Roots (**) D - ri = 0 is clled the chrcteristic eqution of D nd x is clled chrcteristic vector of D. By definition, n x is not unique. If [D - ri]x = 0, then [D - ri]kx = 0, for ny k. To remove the indetermincy, x is normlized so tht x'x = 1. Chrcteristic Roots (**) D - ri = 0 represents n n th degree polynomil l in r which h hs n roots. If D is symmetric, then these roots re rel numbers. 39

40 An Exmple D = [D - ri] = 2 r r Solution r 2 - r - 6 = 0. r 1, r 2 = 1 ( ) / = 1/2 5/2 = 3, Recll tht the solution to x 2 + bx + c = 0 is x 1,x 2 = (-b (b 2-4c) 1/2 )/2 40

41 Solution [D - ri] x x 1 2 = x x 1 2 = 0 0 -x 1 + 2x 2 = 0 2x 1-4x 2 = 0. Solution Eqution 1 is just multiple of eqution 2. They re not independent s expected. All tht we cn conclude from these equtions is tht (1) x 1 = 2x 2. If we impose the normliztion constrint (2) x 12 + x 22 = 1, then (1) nd (2) give us two equtions in two unknowns. Solving (1) nd substituting (2x 2 ) 2 + x 22 = 1 x 1 = 2/(5) 1/2 nd x 2 = 1/(5) 1/2. 41

42 Solution The chrcteristic vector is v 1 = (v 11, v 21 ) = (2/(5) 1/2, 1/(5) 1/2 ). Using the sme technique for r 2 = -2, we cn show tht (x 2 = -2x 1 ) v 2 = (v , v 22 ) = (- 1/(5) 1/2, 2/(5) 1/2 ). Illustrtion 2 x 2 = -2x 1 1 x2 = x1/2 v 2 v

43 Generl Results for Chrcteristic Roots nd Vectors For symmetric mtrix, chrcteristic vectors corresponding to distinct t chrcteristic roots re pirwise orthogonl. v i v i = 1 nd v j v i = 0. If the chrcteristic roots of symmetric mtrix nn re distinct, then they form bsis (orthonorml bsis ) for R n. Generl Results for Chrcteristic Roots nd Vectors The mtrix of chrcteristic vectors of mtrix ti Ai is (v i is n x 1 here) Q = [v 1 v n ]. By definition, Q'Q = I so tht Q' = Q -1. When this condition is met Q is sid to be orthogonl. 43

44 Generl Results for Chrcteristic Roots nd Vectors From the chrcteristic eqution, r AQ = QR, where R r n To see this, note tht AQ = [A v 1 Av n ] = [r 1 v 1 r n v n ] = QR Generl Results for Chrcteristic Roots nd Vectors We conclude tht (*) Q'AQ = Q'QR = R. (*) is clled the digonliztion of A. We hve found mtrix Q such tht the trnsformtion Q'AQ produces digonl mtrix with A's chrcteristic roots long the digonl. 44

45 Generl Results for Chrcteristic Roots nd Vectors For squre mtrix A, we hve i. The product of the chrcteristic roots is equl to the determinte of the mtrix. ii. The rnk of A is equl to the number of nonzero chrcteristic roots. iii. The chrcteristic roots of A 2 re the squres of the chrcteristic roots A, but the chrcteristic vectors of both mtrices re the sme. iv. The chrcteristic roots of A -1 re the reciprocl of the chrcteristic roots of A, but the chrcteristic vectors of both mtrices re the sme. Generl Results on the Trce of Mtrix tr(ca) = c(tr(a)). tr(a') = tr(a). tr(a+b) = tr(a) + tr(b). tr(i k ) = k. tr(ab) = tr(ba). (Note this cn be extended to ny permuttion: tr(abcd) = tr(bcda) = tr(cdab) = tr(dabc). 45

46 Qudrtic Forms A qudrtic form is homogeneous polynomil of the second degree. It tkes on the form x'ax, where A is symmetric nd n by n, nd x is n by 1. We hve x'ax = ij x i x j. x'ax is termed negtive definite if it is negtive for ll x 0 n. The form nd the mtrix re termed negtive definite in this cse. Qudrtic Forms The definitions for positive definite re nlogous with the inequlity sign reversed. Next we consider new concept defined s the principl minor of n n x n mtrix A: PM i is the determinnt of the submtrix of A formed by retining only the first i rows nd columns of A. 46

47 Exmple of PM i Exmple PM PM PM3 A Opertionl Tests for Definite Qudrtic Forms Proposition 1. A nd its qudrtic form re negtive definite it if nd only if principl i minors of order i re of sign (-1) i. Is the following definite? 1 1/2 1/

48 Opertionl Tests for Definite Qudrtic Forms Proposition 2. A nd its qudrtic form re positive definite it if nd only if principl i minors of order i re of positive sign. Exmple: Is the following definite?

49 Exmple If A is 2 x 2 nd negtive definite, is it true tht t 22 < 0? If A is 2 x 2 nd positive definite, is it true tht 22 > 0? Other Opertionl Tests for Definite Qudrtic Forms nd Mtrices Another equivlent condition is given in Proposition 3. A mtrix A is negtive (positive) definite if nd only if ll of its chrcteristic roots re negtive (positive). Remrk: Semidefinite mtrices re Remrk: Semidefinite mtrices re defined s bove with replcing >. 49

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting Self-Invertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

4 Approximations. 4.1 Background. D. Levy

4 Approximations. 4.1 Background. D. Levy D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by

2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by Teoretisk Fysik KTH Advnced QM SI2380), Exercise 8 12 1. 3 Consider prticle in one dimensions whose Hmiltonin is given by Ĥ = ˆP 2 2m + V ˆX) 1) with [ ˆP, ˆX] = i. By clculting [ ˆX, [ ˆX, Ĥ]] prove tht

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Unambiguous Recognizable Two-dimensional Languages

Unambiguous Recognizable Two-dimensional Languages Unmbiguous Recognizble Two-dimensionl Lnguges Mrcell Anselmo, Dor Gimmrresi, Mri Mdoni, Antonio Restivo (Univ. of Slerno, Univ. Rom Tor Vergt, Univ. of Ctni, Univ. of Plermo) W2DL, My 26 REC fmily I REC

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Chapter 04.05 System of Equations

Chapter 04.05 System of Equations hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

Vector differentiation. Chapters 6, 7

Vector differentiation. Chapters 6, 7 Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

19. The Fermat-Euler Prime Number Theorem

19. The Fermat-Euler Prime Number Theorem 19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Linear Equations in Two Variables

Linear Equations in Two Variables Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1

More information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 POSITIVE INTEGRAL EXPONENTS 54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

More information

Math 312 Homework 1 Solutions

Math 312 Homework 1 Solutions Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5. . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

Lecture 25: More Rectangular Domains: Neumann Problems, mixed BC, and semi-infinite strip problems

Lecture 25: More Rectangular Domains: Neumann Problems, mixed BC, and semi-infinite strip problems Introductory lecture notes on Prtil ifferentil Equtions - y Anthony Peirce UBC 1 Lecture 5: More Rectngulr omins: Neumnn Prolems, mixed BC, nd semi-infinite strip prolems Compiled 6 Novemer 13 In this

More information

1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011 - Final Exam

1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011 - Final Exam 1./1.1 Introduction to Computers nd Engineering Problem Solving Fll 211 - Finl Exm Nme: MIT Emil: TA: Section: You hve 3 hours to complete this exm. In ll questions, you should ssume tht ll necessry pckges

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information

Karlstad University. Division for Engineering Science, Physics and Mathematics. Yury V. Shestopalov and Yury G. Smirnov. Integral Equations

Karlstad University. Division for Engineering Science, Physics and Mathematics. Yury V. Shestopalov and Yury G. Smirnov. Integral Equations Krlstd University Division for Engineering Science, Physics nd Mthemtics Yury V. Shestoplov nd Yury G. Smirnov Integrl Equtions A compendium Krlstd Contents 1 Prefce 4 Notion nd exmples of integrl equtions

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 2010-2011

M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 2010-2011 M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 21-211 1. Clculte the men, vrince nd chrcteristic function of the following probbility density functions. ) The exponentil distribution

More information

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES Solution to exm in: FYS30, Quntum mechnics Dy of exm: Nov. 30. 05 Permitted mteril: Approved clcultor, D.J. Griffiths: Introduction to Quntum

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

AA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson

AA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson AA1H Clculus Notes Mth1115, Honours 1 1998 John Hutchinson Author ddress: Deprtment of Mthemtics, School of Mthemticl Sciences, Austrlin Ntionl University E-mil ddress: John.Hutchinson@nu.edu.u Contents

More information

2.016 Hydrodynamics Prof. A.H. Techet

2.016 Hydrodynamics Prof. A.H. Techet .01 Hydrodynics Reding #.01 Hydrodynics Prof. A.H. Techet Added Mss For the cse of unstedy otion of bodies underwter or unstedy flow round objects, we ust consider the dditionl effect (force) resulting

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Notes on Symmetric Matrices

Notes on Symmetric Matrices CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6 Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

1B METHODS LECTURE NOTES. PART I: Fourier series, Self adjoint ODEs

1B METHODS LECTURE NOTES. PART I: Fourier series, Self adjoint ODEs 1B Methods 1. 1B METHODS ECTURE NOTES Richrd Jozs, DAMTP Cmbridge rj31@cm.c.uk October 213 PART I: Fourier series, Self djoint ODEs 1B Methods 2 PREFACE These notes (in four prts cover the essentil content

More information

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010 /28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Solution to Problem Set 1

Solution to Problem Set 1 CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

BUSINESS MATHEMATICS

BUSINESS MATHEMATICS BUSINESS MATHEMATICS HIGHER SECONDARY - SECOND YEAR olume- Untouchbility is sin Untouchbility is crime Untouchbility is inhumn TAMILNADU TEXTBOOK CORORATION College Rod, Chenni - 6 6. Government of Tmilndu

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information