# LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

Size: px
Start display at page:

Transcription

1 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 liner trnsformtions 8 LINEAR TRANSFORMATIONS DEFINITION A function T : R n R m is liner if it stisfies two properties: () For ny vectors v nd w in R n, (2) T (v + w) T (v) + T (w) ( comptibility with ddition ); (2) For ny vector v R n nd ny λ R, (3) T (λv) λt (v) ( comptibility with sclr multipliction ) Often liner mpping T : R n R m is clled liner trnsformtion or liner mp Some uthors reserve the word function for rel-vlued function, nd cll vector-vlued mpping trnsformtion or mpping In the specil cse when n m bove, so the domin nd trget spces re the sme, we cll liner mp T : R n R n liner opertor From the two properties () nd (2) defining linerity bove, number of fmilir fcts follow For instnce: 4 PROPOSITION A liner trnsformtion crries the origin to the origin Tht is, if T : R n R m is liner nd k R k denotes the zero vector in R k, then T ( n ) m Proof Consider the eqution n + n n in R n Apply T to both sides: then in R m we hve the eqution T ( n + n ) T ( n ) By the comptibility of T with ddition (the first property bove), this becomes T ( n ) + T ( n ) T ( n ) Now dd T ( n ) to both sides to obtin T ( n ) m 5 EXAMPLES We consider some exmples nd some non-exmples of liner trnsformtions () The identity mp Id : R n R n defined by Id(v) v is liner (check this) (2) Define R : R 3 R 3 by R x y x y, z z Dte: September 7, 2

2 2 DAVID WEBB reflection through the xy-plne Then R is liner Indeed, R v v 2 + w w 2 R v + w v 2 + w 2 v + w v 2 + w 2 v 3 w 3 v 3 + w 3 (v 3 + w 3 ) v v 2 + w w 2 R v v 2 + R w w 2, v 3 w 3 v 3 w 3 so (2) holds, nd one checks (3) similrly ([ [ x x (3) The mpping T : R 2 R 2 defined by T is liner (check this) y]) x] ([ ([ x x (4) The mpping T : R 2 R 2 given by T is not liner: indeed, T y]) y ]) but we know by Proposition (4) tht if([ T were liner, then we would hve T () x x (5) The mpping T : R 2 R 2 given by T y]) x 2 is not liner (why not?), (6) Any liner mpping T : R R is given by T (x) mx for some m R; ie, its grph in R 2 is line through the origin To see this, let m T () Then by (3), T (x) T (x ) xt () xm mx Wrning: Note tht liner mp f : R R in the sense of Definition () is not wht you probbly clled liner function in single-vrible clculus; the ltter ws function of the form f(x) mx + b whose grph need not pss through the origin The proper term for function like f(x) mx + b is ffine (Some uthors cll such function ffine liner, but the terminology is misleding it incorrectly suggests tht n ffine liner mp is specil kind of liner mp, when in fct s noted bove, n ffine mp is usully not liner t ll! We will use the term ffine in order to void this potentil confusion More generlly, n ffine mp A : R n R m is mpping of the form A(x) M(x) + b for x R n, where M : R n R m is liner mp nd b R m is some fixed vector in the trget spce) (7) (Very importnt exmple) Let A be ny m n mtrix Then A defines liner mp L A : R n R m ( left-multipliction by A ) defined by (6) L A (v) Av for v R n Note tht this mkes sense: the mtrix product of A (n n m mtrix) nd v ( column vector in R n, ie, n n mtrix) is n m mtrix, ie, column vector in R m Tht L A is liner is cler from some bsic properties of mtrix multipliction For exmple, to check tht (2) holds, note tht for v, w R n, L A (v + w) A(v + w) Av + Aw L A (v) + L A (w), nd (3) is checked similrly As specil cse of this lst exmple, let A Then () [ x x L A y ] y [ x x], so in this cse L A is just the liner function T of the third exmple bove

3 LINEAR MAPS, REPRESENTING MATRICES 3 2 THE REPRESENTING MATRIX OF A LINEAR TRANSFORMATION The remrkble fct tht mtrix clculus so useful is tht every lienr trnsformtion T : R n R m is of the form L A for some suitble m n mtrix A; this mtrix A is clled the representing mtrix of T, nd we my denote it by [T ] (2) Thus A [T ] is just nother wy of writing T L A At the risk of belboring the obvious, note tht sying tht mtrix A is the representing mtrix of liner trnsformtion T just mounts to sying tht for ny vector v in the domin of T, T (v) Av (the product on the right side is mtrix multipliction), since T (v) L A (v) Av by definition of L A Proof To see tht every liner trnsformtion T : R n R m hs the form L A for some m n mtrix A, let s consider the effect of T on n rbitrry vector in the domin R n Let x x v 2 x n e, e 2,, e n be the stndrd coordinte unit vectors (in R 3, these vectors re trditionlly denoted i, j, k) Then x x v 2 x + x x n x e + x 2 e x n e n, x n so T (v) T (x e + x 2 e x n e n ) T (x e ) + T (x 2 e 2 ) + + T (x n e n ) (22) x T (e ) + x 2 T (e 2 ) + + x n T (e n ) Thus, to know wht T (v) is for ny v, ll we need to know is the vectors T (e ), T (e 2 ),, T (e n ) Ech of these is vector in R m Let s write them s 2 n T (e ) 2, T (e 2) 22,, T (e n) 2n ; m m2 mn

4 4 DAVID WEBB thus ij is the ith component of the vector T (e j ) R m By (22), where T (v) x T (e ) + x 2 T (e 2 ) + + T (e n ) x 2 + x x 2n n m m2 mn x + 2 x n x n 2 n x 2 x + 22 x n x n n x 2 Av L A(v), m x + m2 x mn x n m m2 mn x n 2 n A n m m2 mn Thus we hve shown tht T is just L A, where A is the m n mtrix whose columns re the vectors T (e ), T (e 2 ),, T (e n ); equivlently: (23) The representing mtrix [T ] is the mtrix whose columns re the vectors T (e ), T (e 2 ),, T (e n ) 2 n We might write this observtion schemticlly s [T ] T (e ) T (e 2 ) T (e n ) 24 EXAMPLE Let us determine the representing mtrix [Id] of the identity mp Id : R n R n By (23), the columns of [Id] re Id(e ), Id(e 2 ),, Id(e n ), ie, e, e 2,, e n But the mtrix whose columns re e, e 2,, e n is just, the n n identity mtrix I Thus [Id] I Note tht two liner trnsformtions T : R n R m nd U : R n R m tht hve the sme representing mtrix must in fct be the sme trnsformtion: indeed, if [T ] A [U], then by (2), T L A U Finlly, there is wonderfully convenient fct tht helps to orgnize mny seemingly complicted computtions very simply; for exmple, s we will see lter, it permits very simple nd esily remembered sttement of the multivrible Chin Rule Suppose tht T : R n R m nd U : R m R p re liner trnsformtions Then there is composite liner trnsformtion U T : R n R p given by (U T )(v) U(T (v)) for v R n

5 LINEAR MAPS, REPRESENTING MATRICES 5 R n T R m U R p U T 25 THEOREM [U T ] [U] [T ] Tht is, the representing mtrix of the composite of two liner trnsformtions is the product of their representing mtrices (Note tht this mkes sense: [T ] is n m n mtrix while U is p m mtrix, so the mtrix product [U][T ] is p n mtrix, s [U T ] should be if it represents liner trnsformtion R n R p ) Proof To see why this is true, let A [T ] nd B [U] By (2), this is just nother wy of sying tht T L A nd U L B Then for ny vector v R n, (U T )(v) U(T (v)) L B (L A (v)) B(Av) (BA)v L BA (v), where in the fourth equlity we hve used the ssocitivity of mtrix multipliction Thus U T L BA, which by (2) is just nother wy of sying tht [U T ] BA, ie, tht [U T ] [U] [T ] In fct, this theorem is the reson tht mtrix multipliction is defined the wy it is! 26 EXAMPLE Consider the liner trnsformtion R θ tht rottes the plne R 2 counterclockwise by n ngle θ Clerly, () cos θ R θ (e ) R θ, sin θ by definition of the sine nd cosine [ functions ] Now e 2 is perpendiculr to e, so R θ (e 2 ) should be unit cos θ vector perpendiculr to R θ (e ), ie, R sin θ θ (e 2 ) is one of the vectors sin θ sin θ, cos θ cos θ From the picture it is cler tht R θ (e 2 ) sin θ, cos θ since the other possibility would lso involve reflection By (23), the representing mtrix [R θ ] of R θ is the mtrix whose columns re R θ (e ) nd R θ (e 2 ); thus we hve cos θ sin θ (27) [R θ ] sin θ cos θ Consider nother rottion R ϕ by n ngle ϕ; its representing mtrix is given by cos ϕ sin ϕ [R ϕ ] sin ϕ cos ϕ Similrly, the representing mtrix of the rottion R ϕ+θ through the ngle ϕ + θ is given by cos(ϕ + θ) sin(ϕ + θ) [R ϕ+θ ] sin(ϕ + θ) cos(ϕ + θ) Now the effect of rotting by n ngle θ nd then by n ngle ϕ should be simply rottion by the ngle ϕ+θ, ie, (28) R ϕ+θ R ϕ R θ

6 6 DAVID WEBB Thus By Theorem (25), this becomes ie, cos(ϕ + θ) sin(ϕ + θ) sin(ϕ + θ) cos(ϕ + θ) [R ϕ+θ ] [R ϕ R θ ] [R ϕ+θ ] [R ϕ ] [R θ ], [ cos ϕ sin ϕ cos θ sin θ sin ϕ cos ϕ sin θ cos θ After multiplying out the right hnd side, this becomes cos(ϕ + θ) sin(ϕ + θ) cos ϕ cos θ sin ϕ sin θ cos ϕ sin θ sin ϕ cos θ sin(ϕ + θ) cos(ϕ + θ) sin ϕ cos θ + cos ϕ sin θ sin ϕ sin θ + cos ϕ sin θ Compring the entries in the first column of ech mtrix, we obtin (29) (2) cos(ϕ + θ) cos ϕ cos θ sin ϕ sin θ, sin(ϕ + θ) sin ϕ cos θ + cos ϕ sin θ Thus from (28) long with Theorem (25) we recover the trigonometric sum identities 3 AN APPLICATION: REFLECTIONS IN THE PLANE Let l θ be line through the origin in the plne; sy its eqution is y mx, where m tn θ; thus θ is the ngle of tht the line l θ mkes with the x-xis Consider the liner trnsformtion F θ : R 2 R 2 clled reflection through l θ defined geometriclly s follows: intuitively, we view the line l θ s mirror; then for ny vector v R 2 (viewed s the position vector of point P in the plne), F θ (v) is the position vector of the mirror imge point of P on the other side of the line l θ More precisely, write v s sum of two vectors: (3) v v + v, where v is vector long the line l θ nd v is perpendiculr to l θ Then F θ (v) v v We wish to determine n explicit formul for F θ This cn be done using elementry plne geometry nd fmilir fcts bout the dot product, but we seek here to outline method tht works in high dimensions s well, when it is not so esy to visulize the mp geometriclly To this end, note tht we lredy know formul for F θ when θ : indeed, in tht cse, the line l θ is just the x-xis, nd the reflection F is given by () [ x x F y y] For our purposes, though, more useful (but equivlent) wy of sying this is vi the representing mtrix [F ], which we now compute The representing mtrix [F ] is the mtrix whose columns re F (e ) nd F (e 2 ) It is obvious tht F (e ) e nd F (e 2 ) e 2, but let s check it nywy It we tke v e in eqution (3), we see tht v e (v e is on the line l, the x-xis), so v, the zero vector; thus F (e ) v v e e Similrly, if we tke v e 2 in eqution (3), we find v while v e 2 (v e 2 is itself perpendiculr to the x-xis), so F (e 2 ) v v e 2 e 2 Thus [F ] is the mtrix whose columns re e nd e 2, ie, (32) [F ] ]

7 LINEAR MAPS, REPRESENTING MATRICES 7 To use this seemingly trivil observtion to compute F θ, we note tht F θ cn be written s composition of three liner trnsformtions: (33) F θ R θ F R θ, where R θ is the rottion by θ opertor whose representing mtrix we lredy determined in eqution (27) Tht is, in order to reflect through the line l θ, we cn proceed s follows: first, rotte the whole plne through n ngle θ; this opertion moves the mirror line l θ to the x-xis We now reflect the plne through the x-xis, which we know how to do by eqution (32) Finlly, we rotte the whole plne bck through n ngle of θ in order to undo the effect of our originl rottion through n ngle of θ; this opertion restores the line in which we re interested (the mirror ) to its originl position, mking n ngle θ with the x-xis By Theorem (25), we cn now compute [F θ ] esily: [F θ ] [R θ F R θ ] [R θ ][F ][R θ ], ie, cos θ sin θ cos( θ) sin( θ) [F θ ] sin θ cos θ sin( θ) cos( θ) multiplying out the mtrices yields [F θ ] [ cos θ sin θ sin θ cos θ cos 2 θ sin 2 θ 2 cos θ sin θ 2 cos θ sin θ sin 2 θ cos 2 θ Using the identities (29) with ϕ θ, we finlly conclude tht cos(2θ) sin(2θ) (34) [F θ ] sin(2θ) cos(2θ) ] [ ] cos θ sin θ ; sin θ cos θ To see tht this mkes some sense geometriclly, note tht the first column of the mtrix [F θ ] is the sme s the first column of [R 2θ ], the representing mtrix of the rottion through n ngle 2θ But for ny liner mp T : R 2 R 2, the first column of [T ] is just the vector T (e ) Thus the equlity of the first columns of [F θ ] nd [R 2θ ] just mens tht F θ (e ) R 2θ (e ), which mkes geometric sense: if you reflect the horizontl vector e through the line l θ, the resulting vector should surely be unit vector mking n ngle θ with the line l θ, hence mking n ngle 2θ with the x-xis Although we will not mke serious use of the striking power of the technique illustrted bove, we conclude by showing how the lgebr of liner trnsformtions, once codified vi their representing mtrices, cn led to illuminting geometric insights We consider very simple exmple to illustrte Let us rewrite eqution (34): cos(2θ) sin(2θ) cos(2θ) sin(2θ) [F θ ] [F θ ], sin(2θ) cos(2θ) sin(2θ) cos(2θ) the right hnd side of which we immeditely recognize s [R 2θ ][F ] But by Theorem (25), the ltter is just [R 2θ F ] Thus [F θ ] [R 2θ F ] Since two liner trnsformtions with the sme representing mtrix re the sme, it follows tht F θ R 2θ F Now compose on the right by F nd use ssocitivity of function composition: (35) F θ F (R 2θ F ) F R 2θ (F F )

8 8 DAVID WEBB Finlly, observe tht F F Id, the identity trnsformtion (This is obvious geometriclly, but we cn prove it crefully by using the representing mtrix: by Theorem (25) nd eqution (32), [F F ] [F ][F ] I [Id], so F F Id) Then eqution (35) becomes (36) F θ F R 2θ Thus we hve proved: 37 THEOREM The effect of reflection of the plne through the x-xis followed by reflection of the plne through the line l θ mking n ngle θ with the x-xis is just tht of rottion of the plne through the ngle 2θ You should drw pictures nd convince yourself of this fct In fct, more is true: for ny line λ in the plne, let F λ : R 2 R 2 denote reflection through the line λ 38 THEOREM For ny two lines α nd β in the plne, the composite F β F α is the rottion through double the ngle between the lines α nd β The proof is immedite: we merely set up our coordinte xes so tht the x-xis is the line α; then we hve reduced to the cse of Theorem (37) 4 THE ALGEBRA OF LINEAR TRANSFORMATIONS We will denote the collection of ll liner mps T : R n R m by L(R n, R m ) We cn dd two elements of L(R n, R m ) in n obvious wy: 4 DEFINITION If T, U L(R n, R m ), then T + U is the mpping given by (T + U)(v) T (v) + U(v) for ll vectors v R n You should check tht T + U is indeed liner 42 THEOREM Let T, U L(R n, R m ) Then [T + U] [T ] + [U] In words, the representing mtrix of the sum of two liner trnsformtions is the sum of their representing mtrices The proof is simple exercise using the definition of the representing mtrix nd the definition of mtrix ddition 43 DEFINITION Given liner mp T : R n R m nd sclr c R, we define liner mp ct from R n to R m (clled sclr multipliction of T by c) by (ct )(v) c T (v) for ll v R n As bove, you should check tht ct is liner, nd prove the following theorem: 44 THEOREM Let T L(R n, R m ), nd let c R Then [ct ] c[t ] Note tht the two definitions bove endow the set L(R n, R m ) with the structure of vector spce The definition of mtrix ddition nd sclr multipliction endow the spce M m n (R) of m by n mtrices (with rel entries) with the structure of vector spce There is mpping : L(R n, R m ) M m n (R) tht sends liner trnsformtion T to its representing mtrix [T ] The two simple theorems bove merely sy tht this mpping is itself liner mp, using the vector spce structures on its domin nd trget tht we

9 LINEAR MAPS, REPRESENTING MATRICES 9 defined bove! But is even better thn just liner mp: it hs n inverse L ( ) : M m n (R) L(R n, R m ) tht sends mtrix A M m n (R) to the liner trnsformtion L A : R n R m given by L A (v) Av The fct tht nd L ( ) re inverses of ech other is just the observtion (2) Since : L(R n, R m ) M m n is liner, ny liner question bout liner mps from R n to R m (ie, ny question involving ddition of such liner mps or multipliction by sclr) cn be trnslted vi to question bout mtrices, which re much more concrete nd better suited to computtion thn bstrct entities like liner trnsformtions; the fct tht hs liner inverse L ( ) mens tht nothing is lost in the trnsltion The mpping is clssic exmple of n isomorphism Formlly, n isomorphism of two vector spces V nd W is just liner mp V W with liner inverse W V The ide is tht V nd W re to ll intents nd purposes the sme, t lest s fr s liner lgebr is concerned An isomorphism of n bstrct vector spce V with more concrete nd redily understood vector spce W mkes V just s esy to understnd s W : we simply use the isomorphism to trnslte question bout V to n equivlent question bout W, solve our problem in W (where is is esy), then use the inverse of the isomorphism to trnslte our solution bck to V While we will not mke serious use of the ide of isomorphism (t lest not explicitly), there is one importnt exmple tht will use, nd in ny cse the ide is so importnt throughout mthemtics tht is is worth mking it explicit t this point We will show tht the spce of liner mps L(R, R m ) of liner mps from the rel line to R m is the sme s the spce R m itself In fct, ny liner mp T : R R n hs representing mtrix [T ] M m in the spce of m by mtrices; but n m by mtrix (m rows, one column) is just column vector of length m, ie, n element of R m Wht does this representing mtrix [T ] look like? Recll tht its columns re just the vlues tht the liner mp T tkes on the stndrd coordinte vectors But in R, there is only one stndrd coordinte vector: e, the rel number Thus the first (nd only!) column of [T ] is just the vector T () R m We cn summrize s follows: (45) We cn view ny liner mp T : R R m s column vector in R m ; we do so by ssociting to the liner mp T the vector T () R m which, when viewed s n m by mtrix, is just the representing mtrix [T ] Another wy of rriving t the sme conclusion is the following If T : R R m is ny liner mp from the rel line to R m, note tht for ny x R, T (x) T (x ) xt (), by (3); equivlently, T (x) T ()x, which just sys tht T L T (), ie (by (2)), [T ] T ()

### 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

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.................................................

### 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

### 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

### 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

### 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

### . 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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 +

### 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.

### 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

### 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

### 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

### 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

### 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

### 6.2 Volumes of Revolution: The Disk Method

mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

### Basic Analysis of Autarky and Free Trade Models

Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

### Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

### Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

### 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

### 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

### 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

### 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

### 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

### 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:

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

### 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

### Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.

Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from

### 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

### 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

### Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems

Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch

### Exponential and Logarithmic Functions

Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

### 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

### Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

### 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

### Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

### Words Symbols Diagram. abcde. a + b + c + d + e

Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

### addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

### 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

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### 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.

### 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

### Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

### 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}.

### 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.

### PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

### 15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

### Clipping & Scan Conversion. CSE167: Computer Graphics Instructor: Steve Rotenberg UCSD, Fall 2005

Clipping & Scn Conersion CSE167: Computer Grphics Instructor: Stee Rotenberg UCSD, Fll 2005 Project 2 Render 3D hnd (mde up of indiidul boxes) using hierrchicl trnsformtions (push/pop) The hnd should perform

### 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

### Rotating DC Motors Part II

Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

### 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

### 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

### GENERALIZED QUATERNIONS SERRET-FRENET AND BISHOP FRAMES SERRET-FRENET VE BISHOP ÇATILARI

Sy 9, Arlk 0 GENERALIZED QUATERNIONS SERRET-FRENET AND BISHOP FRAMES Erhn ATA*, Ysemin KEMER, Ali ATASOY Dumlupnr Uniersity, Fculty of Science nd Arts, Deprtment of Mthemtics, KÜTAHYA, et@dpu.edu.tr ABSTRACT

### Repeated multiplication is represented using exponential notation, for example:

Appedix A: The Lws of Expoets Expoets re short-hd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you

### CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

### 6 Energy Methods And The Energy of Waves MATH 22C

6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this

### 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

### 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

### 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

### 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

### 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

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

### 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

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e

### Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I

Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

### 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