LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

Size: px
Start display at page:

Download "LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES"

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 ()

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

More information

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

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

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

has the desired form. On the other hand, its product with z is 1. So the inverse x

has the desired form. On the other hand, its product with z is 1. So the inverse x First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins

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

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

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

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

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

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

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

Notes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams

Notes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly

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

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

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

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

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

Quadratic Functions. Analyze and describe the characteristics of quadratic functions

Quadratic Functions. Analyze and describe the characteristics of quadratic functions Section.3 - Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Addition and subtraction of rational expressions

Addition and subtraction of rational expressions Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The

More information

Scalar Line Integrals

Scalar Line Integrals Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π. . Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

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

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

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

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

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic Equations. Math 99 N1 Chapter 8 Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

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

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

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

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

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

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

Lecture 2: Matrix Algebra. General

Lecture 2: Matrix Algebra. General 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

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

1. Inverse of a tridiagonal matrix

1. Inverse of a tridiagonal matrix Pré-Publicções do Deprtmento de Mtemátic Universidde de Coimbr Preprint Number 05 16 ON THE EIGENVALUES OF SOME TRIDIAGONAL MATRICES CM DA FONSECA Abstrct: A solution is given for problem on eigenvlues

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

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

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

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

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

Section 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that

Section 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the long-promised connection between differentition nd integrtion, between res nd tngent lines.

More information

Exponents base exponent power exponentiation

Exponents base exponent power exponentiation Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily

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

INVERSE OF A MATRIX AND ITS APPLICATION O Q

INVERSE OF A MATRIX AND ITS APPLICATION O Q Inverse f A tri And Its Appliction DUE - III IVERSE F A ATRIX AD ITS AICATI ET US CSIDER A EXAE: Abhinv spends Rs. in buying pens nd note books wheres Shntnu spends Rs. in buying 4 pens nd note books.we

More information

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if

1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should

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

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc. Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

More information

Sect 8.3 Triangles and Hexagons

Sect 8.3 Triangles and Hexagons 13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

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

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

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

6.2 Volumes of Revolution: The Disk Method

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

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

Two special Right-triangles 1. The

Two special Right-triangles 1. The Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function. Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

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

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

Basic Analysis of Autarky and Free Trade Models

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

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Education Spending (in billions of dollars) Use the distributive property.

Education Spending (in billions of dollars) Use the distributive property. 0 CHAPTER Review of the Rel Number System 96. An pproximtion of federl spending on eduction in billions of dollrs from 200 through 2005 cn be obtined using the e xpression y = 9.0499x - 8,07.87, where

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Turing Machine Extensions

Turing Machine Extensions Red K & S 4.3.1, 4.4. Do Homework 19. Turing Mchine Extensions Turing Mchine Definitions An lterntive definition of Turing mchine: (K, Σ, Γ, δ, s, H): Γ is finite set of llowble tpe symbols. One of these

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

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

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

More information

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

The Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.

The Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof. The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

5.1 Second-Order linear PDE

5.1 Second-Order linear PDE 5.1 Second-Order liner PDE Consider second-order liner PDE L[u] = u xx + 2bu xy + cu yy + du x + eu y + fu = g, (x,y) U (5.1) for n unknown function u of two vribles x nd y. The functions,b nd c re ssumed

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

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

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

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

MATLAB Workshop 13 - Linear Systems of Equations

MATLAB Workshop 13 - Linear Systems of Equations MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

More information

Written Homework 6 Solutions

Written Homework 6 Solutions Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f

More information

VECTOR-VALUED FUNCTIONS

VECTOR-VALUED FUNCTIONS VECTOR-VALUED FUNCTIONS MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the book: Sections 13.1. 13.2. Wht students should definitely get: Definition of vector-vlued function, reltion with prmetric

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

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

69. The Shortest Distance Between Skew Lines

69. The Shortest Distance Between Skew Lines 69. The Shortest Distnce Between Skew Lines Find the ngle nd distnce between two given skew lines. (Skew lines re non-prllel non-intersecting lines.) This importnt problem is usully encountered in one

More information

NUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results

NUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results CHAPTER NUMBER SYSTEMS Min Concepts nd Results Rtionl numbers Irrtionl numbers Locting irrtionl numbers on the number line Rel numbers nd their deciml expnsions Representing rel numbers on the number line

More information

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look

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

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

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.

More information

Homework 3 Solution Chapter 3.

Homework 3 Solution Chapter 3. Homework 3 Solution Chpter 3 2 Let Q e the group of rtionl numers under ddition nd let Q e the group of nonzero rtionl numers under multiplition In Q, list the elements in 1 2 In Q, list the elements in

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

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

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information