Chapter 3. Vector Spaces. 3.1 Images and Image Arithmetic

Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner lger, we study suh sets nd develop tools to nlyze them. We will ll these sets vetor spes. 3.1 Imges nd Imge Arithmeti We noted in Chpter 2 tht if you dd two imges, you get new imge, nd tht if you multiply n imge y slr, you get new imge. We sw tht retngulr pixelted imge n e represented s n rry of vlues, or equivlently, s retngulr rry of grysle pthes. This is very nturl ide in the ontext of digitl photogrphy. Rell the definition of n imge given in Chpter 2. We repet it here, nd follow the definition y some exmples of imges with different geometri rrngements. Definition 3.1.1. An imge is finite ordered list of rel vlues with n ssoited geometri rrngement. Four exmples of rrys long with n index system speifying the order of pthes n e seen in Figure 3.1. As n imge, eh pth would lso hve numeril vlue inditing the rightness of the pth (not shown in the figure). The first is regulr pixel rry ommonly used for digitl photogrphy. The seond is hexgon pttern whih lso niely tiles plne. 33

34 CHAPTER 3. VECTOR SPACES The third is mp of the Afrin ontinent nd Mdgsr sudivided y ountry. The fourth is squre pixel set with enhned resolution towrd the enter of the field of interest. It should e ler from the definition tht imges re not mtries. Only the first exmple might e onfused with mtrix. We first fix prtiulr geometri rrngement of pixels (nd let n denote the numer of pixels in the rrngement). Then n imge is preisely desried y its (ordered) intensity vlues. With this determined, we formlize the notions of slr multiplition nd ddition on imges tht were developed in the previous hpter. Definition 3.1.2. Given two imges x nd y with (ordered) intensity vlues (x 1, x 2,, x n ) nd (y 1, y 2,, y n ), respetively, nd the sme geometry; the imge sum, written z x + y is the imge with intensity vlues z i x i + y i for ll i {1, 2,, n}, nd the sme geometry. Hene, the sum of two imges is the imge tht results from pixel-wise ddition of intensity vlues. Put nother wy, the sum of two imges is the imge tht results from dding orresponding vlues of their ordered lists. Definition 3.1.3. Given slr α nd imge x with (ordered) intensity vlues (x 1, x 2,, x n ), the slr produt, written z αx is the imge with intensity vlues z i αx i for ll i {1, 2,, n}, nd the sme geometry. A slr times n imge is the imge tht results from pixel-wise slr multiplition. Tht is, slr times n imge is the imge whih results from multiplition of eh of the vlues in the ordered list y tht slr. We found tht these si rithmeti opertions on imges led to key property: ny omintion of rithmeti opertions on imges results in n imge of the sme onfigurtion. In other words, dding two imges lwys yields n imge, nd multiplying n imge y slr lwys yields n imge. We formlize this notion with the onept of losure:

3.1. IMAGES AND IMAGE ARITHMETIC 35 Figure 3.1: Exmples of imge rrys. Numers indite exmple pixel ordering.

36 CHAPTER 3. VECTOR SPACES Definition 3.1.4. Consider set of ojets X with slrs tken from R, nd opertions of ddition (+) nd slr multiplition ( ) defined on X. We sy tht X is losed under ddition if x + y X for ll x, y X. We sy tht X is losed under slr multiplition if α x X for eh x X nd eh α R. Let I m n denote the set of ll m n retngulr imges. We see tht the set I 4 4 (imges used in Chpter 2) is losed under ddition nd slr multiplition. This rithmeti with imges in I 4 4 lso stisfies numer of other nturl properties: (Commuttivity of imge ddition.) If I A nd I B re imges in I 4 4, then I A + I B I B + I A. For exmple, + +. Imge A Imge B Imge B Imge A (Assoitivity of imge ddition.) If I A, I B, nd I C re imges in I 4 4, then (I A + I B ) + I C I A + (I B + I C ). For exmple, + + + +. Imge A Imge B Imge C Imge A Imge B Imge C (Assoitivity of slr multiplition.) If α, β R nd I I 4 4, then α (β I) (αβ) I, e.g., 7 3 21. Imge A Imge A (Distriutivity of Slr Multiplition over Imge Addition) If α R nd I A, I B I 4 4, then α (I A + I B ) α I A + α I B, e.g., 7 + 7 + 7. Imge A Imge B Imge A Imge B

3.2. VECTORS AND VECTOR SPACES 37 (Additive identity imge.) There is zero imge in I 4 4 the imge tht hs every pixel intensity equl to zero. The sum of the zero imge nd ny other imge I is just I. (Additive inverses.) For every imge I I 4 4, there is n imge J so tht the sum I +J is just the zero imge. (Rell tht the set of imges inlude those tht n e ptured y your mer, ut there re mny more, some with negtive pixel intensities s well.) (Multiplitive identity.) For ny imge I I 4 4 the slr produt 1 I I. The ft tht the spe I 4 4 of 4 4 hs these properties will enle us to develop tools for working with imges. In ft, we will e le to develop tools for ny set (nd field of slrs) tht stisfies these properties. We will ll suh sets vetor spes. 3.2 Vetors nd Vetor Spes In the lst setion, we sw tht the set of 4 4 imges, together with rel slrs, stisfies severl nturl properties. There re in ft mny other sets of ojets tht lso hve these properties. One lss of ojets with these properties re the vetors tht you my hve seen in ourse in multivrile lulus or physis. In those ourses, vetors re ojets with fixed numer, sy n, of vlues put together into n ordered tuple. Tht is, the word vetor my ring to mind something tht looks like,,,,, or 1, 2,..., n. Mye you ve even seen things like ny of the following (, ), (,, ), ( 1, 2,..., n ), lled vetors s well. In this setion, we generlize the notion of vetor. In prtiulr, we will understnd tht imges nd other lsses of ojets n e vetors in n pproprite ontext. When we onsider ojets like rin imges, rdiogrphs or het stte signtures, it is often useful to understnd them s,, 1 2. n, 1 2. n

38 CHAPTER 3. VECTOR SPACES olletions hving ertin nturl mthemtil properties. Indeed, we will develop mthemtil tools tht n e used on ll suh sets, nd these tools will e instrumentl in omplishing our pplition tsks. We hven t yet mde the definition of vetor spe (or even vetor) rigorous. We still hve some more setup to do. In this text, we will use two slr fields 1 :R nd Z 2. Z 2 is the two element (or inry) set {0, 1} with ddition nd multiplition omputed modulo 2. Here, ddition omputed modulo 2, mens tht: 0 + 0 0, 0 + 1 1 + 0 1, nd 1 + 1 0. Multiplition modulo 2 works s we expet: 0 0 0, 0 1 1 0 0, nd 1 1 1. We n think of the two elements s on nd off nd the opertions s inry opertions. If we dd 1, we flip the swith nd if we dd 0, we do nothing. Notie tht Z 2 is losed under slr multiplition nd vetor ddition. 1 The definition of field n e found in Appendix F. The importnt thing to rememer out fields (for this liner lger ourse) is tht there re two opertions (typilly ddition nd multiplition) tht stisfy properties we usully see with rel numers.

3.2. VECTORS AND VECTOR SPACES 39 Definition 3.2.1. Consider set V over field F (either R or Z 2 ) with given definitions for ddition (+) nd slr multiplition ( ). V with + nd is lled vetor spe over F if for ll u, v, w V nd for ll α, β F, the following ten properties hold. (P1) Closure Property for Addition u + v V. (P2) Closure Property for Slr Multiplition α v V. (P3) Commuttive Property for Addition u + v v + u. (P4) Assoitive Property for Addition (u+v)+w u+(v +w). (P5) Assoitive Property for Slr Multiplition α (β v) (αβ) v. (P6) Distriutive Property of Slr Multiplition Over Vetor Addition α (u + v) α u + α v. (P7) Distriutive Property of Slr Multiplition Over Slr Addition (α + β) v α v + β v. (P8) Additive Identity Property V ontins the dditive identity, denoted 0 so tht 0 + v v + 0 v. (P9) Additive Inverse Property V ontins dditive inverses z so tht v + z 0. (P10) Multiplitive Identity Property for Slrs The slr set F hs n identity element, denoted 1, for slr multiplition tht hs the property 1 v v. Note: In Definition 3.2.1, we lel the properties s (P1)-(P10). It should e noted tht though we use this leling in other ples in this text, these lels re not stndrd. This mens tht the reder should fous more on the property desription nd nme rther thn the leling.

40 CHAPTER 3. VECTOR SPACES Definition 3.2.2. Given vetor spe (V, +, ) over F. We sy tht v V is vetor. Tht is, elements of vetor spe re lled vetors. Note: In this text, we will use ontext to indite vetors with letter, suh s v or x. In some ourses nd textooks, vetors re denoted with n rrow over the nme, v, or with old type, v. We will disuss mny vetor spes for whih there re opertions nd fields tht re typilly used. For exmple, the set of rel numers R is set for whih we hve ommon understnding out wht it mens to dd nd multiply. For these vetor spes, we ll the opertions the stndrd opertions nd the field is lled the stndrd field. So, we might typilly sy tht R, with the stndrd opertions, is vetor spe over itself. Wth Your Lnguge! Notie tht the lnguge used to speify vetor spe requires tht we stte the set V, the two opertions + nd, nd the field F. We mke this ler with nottion nd/or in words. Two wys to ommunite this re (V, +, ) is vetor spe over F. or V with the opertions + nd is vetor spe over the field F. We should not sy (unless miguity hs een removed) V is vetor spe. Definition 3.2.1 is so importnt nd hs so mny piees tht we will tke the time to present mny exmples in this hpter nd the next hpter. As we do so, onsider the following. The identity element for slr multiplition need not e the numer 1. The zero vetor need not e (nd in generl is not) the numer 0. The elements of vetor spe re lled vetors ut need not look like the vetors presented ove.

3.2. VECTORS AND VECTOR SPACES 41 Exmple 3.2.1. The set of 4 4 imges I 4 4 stisfies properties (P1)-(P10) of vetor spe. Exmple 3.2.2. Let us onsider the set 1 R 3 2 1, 2, 3 R. 3 This mens tht R 3 is the set of ll ordered triples where eh entry in the triple is rel numer. We n show tht R 3 is lso vetor spe over R with ddition nd slr multiplition defined omponent-wise. This mens tht, for,,, d, f, g, α R, + d f g + d + f + g nd α α α α We show this y verifying tht eh of the ten properties of vetor spe re true for R 3 with ddition nd slr multiplition defined this wy. Proof. Let u, v, w R 3 nd let α, β R.,,, d, f, g, h, k, nd l so tht u, v d f g. Then, there re rel numers, nd w We will show tht properties (P1)-(P10) from Definition 3.2.1 hold. We first show the two losure properties. (P1) Now sine R is losed under ddition, we n sy tht + d, + f, nd + g re rel numers. So, u + v + Thus, R 3 is losed under ddition. d f g + d + f + g h k l. R 3.

42 CHAPTER 3. VECTOR SPACES (P2) Sine R is losed under slr multiplition, we n sy tht α, α, nd α re rel numers. So α α v α α R 3. α Thus, R 3 is losed under slr multiplition. (P3) Now, we show tht the ommuttive property of ddition holds in R 3. Using the ft tht ddition on R is ommuttive, we see tht d + d d + u + v + f + f f + v + u. g + g g + Therefore, ddition on R 3 is ommuttive. (P4) Using the ssoitive property of ddition on R, we show tht ddition on R 3 is lso ssoitive. Indeed, we hve ( + d) + h + (d + h) (u + v) + w ( + f) + k + (f + k) u + (v + w). ( + g) + l + (g + l) So, ddition on R 3 is ssoitive. (P5) Sine slr multiplition on R is ssoitive, we see tht α (β ) (αβ) α (β v) α (β ) (αβ) (αβ) v. α (β ) (αβ) Thus, slr multiplition on R 3 is ssoitive. (P6) Sine property (P6) holds for R, we hve tht α ( + d) α + α d α (u + v) α ( + f) α + α f α ( + g) α + α g α u + α v. Thus, slr multiplition distriutes over vetor ddition for R 3.

3.2. VECTORS AND VECTOR SPACES 43 (P7) Next, using the ft tht (P7) is true on R, we get (α + β) α + β (α + β) v (α + β) (α + β) α + β α + β α v + β v. This mens tht, slr multiplition distriutes over slr ddition for R 3 s well. (P8) Using the dditive identity 0 R, we form the vetor 0 0 0 R 3 0. This element is the dditive identity in R 3. Indeed, 0 + 0 + v 0 +. 0 + Therefore property (P8) holds for R 3. (P9) We know tht,, nd re the dditive inverses in R of,, nd, respetively. Using these, we n form the vetor w R 3. We see tht w is the dditive inverse of v s + ( ) 0 v + w + ( ) + ( ) 0 0 Thus, property (P9) is true for R 3. 0. (P10) Finlly, we use the multiplitive identity, 1 R. Indeed, 1 1 v 1 v. 1 d

44 CHAPTER 3. VECTOR SPACES Now, euse ll ten properties from Definition 3.2.1 hold true for R 3, we know tht R 3 with omponent-wise ddition nd slr multiplition, is vetor spe over R. Notie tht in the ove proof, mny of the properties esily followed from the properties on R nd did not depend on the requirement tht vetors in R 3 were mde of ordered triples. In most ses person would not go through the exruiting detil tht we did in this proof. Beuse the opertions re omponent-wise defined nd the omponents re elements of vetor spe, we n shorten this proof to the following proof. Alternte Proof (for Exmple 3.2.2). Suppose R 3 is defined s ove with ddition nd slr multiplition defined omponent-wise. By definition of the opertions on R 3, the losure properties (P1) nd (P2) hold true. Notie tht ll opertions our in the omponents of eh vetor nd re the stndrd opertions on R. Therefore, sine R with the stndrd opertions is vetor spe over itself, properties (P3)-(P10) re ll inherited from R. Thus R 3 with omponent-wise ddition nd slr multiplition is vetor spe over R. In this proof, we sid,...properties (P3)-(P10) re ll inherited from R to indite tht the justifition for eh vetor spe property for R 3 is just repeted use (in eh omponent) of the orresponding property for R. Exmple 3.2.3. Beuse neither proof relied on the requirement tht elements of R 3 re ordered triples, we see tht very similr proof would show tht for ny n N, R n is vetor spe over the slr field R. Here R n is the set of ordered n-tuples. Cution: There re instnes where vetor spe hs omponents tht re elements of vetor spe, ut not ll elements of this vetor spe re llowed s omponent nd the lternte proof does not work. Exmple 3.2.4. Let T { R 0}. Now, onsider the set 1 T 3 2 1, 2, 3 T 3

3.2. VECTORS AND VECTOR SPACES 45 with ddition nd slr multiplition defined omponent-wise. Notie tht ll omponents of T 3 re rel numers euse ll elements of T re rel numers. But T does not inlude the rel numer 0 nd this mens T 3 is not vetor spe over the field R. Whih property fils? Exerise 3 sks you to nswer this question. Exmple 3.2.4 does not simplify the story either. Cn you think of vetor spe over R, mde of ordered n-tuples, with ddition nd slr multiplition defined omponent wise, whose omponents re in R, nd there re restritions on how the omponents n e hosen? Exerise 7 sks you to explore this nd determine whether or not it is possile. The opertions re themselves very importnt in the definition of vetor spe. Notie tht if we define different opertions on set, the struture of the vetor spe, inluding identity nd inverse elements n e different. Exmple 3.2.5. Let us onsider gin the set of rel numers, R, ut with different opertions. Define the opertion ( ) to e multiplition (u v uv) nd defined to e exponentition (α u u α ). Notie tht is ommuttive ut is not. We show tht (R,, ) is not vetor spe over R. (P1) We know tht when we multiply two rel numers, we get rel numer, thus R is losed under nd this property holds. (P2) However, R is not losed under. For exmple, (1/2) ( 1) ( 1) 1/2 1 is not rel numer. Sine property (P2) does not hold, we do not need to ontinue heking the remining eight properties. To emphsize how the definition of the opertions n hnge vetor spe, we offer more exmples. Exmple 3.2.6. Let V R 2. Let ( nd ) e ( defined ) s inry opertions on V in the following wy. If u, v V nd α R (the d slr set), then u v ( + 1 + d 1 Then, (v,, ) is vetor spe over R. ) nd α u ( α + 1 α α + 1 α ).

46 CHAPTER 3. VECTOR SPACES k Proof. Let u, v, w V nd α, β R nd d l define nd s ove. We will show tht ll 10 properties given in Definition 3.2.1 re true. We egin with the losure properties. (P1) Notie tht sine,,, d R then + 1, + d 1 R. u v R 3. Thus, R 3 is losed under ddition,. So, (P2) Notie lso tht α + 1 α, α + 1 β R, mking α u V. So, V is losed under slr multiplition,. Next, we show the ssoitive nd ommuttive properties. (P3) Notie tht + 1 u v + d 1 + 1 d + 1 v u. Thus, is ommuttive opertion. (P4) We show the ssoitive property of, using the ssoitivity nd ommuttivity of ddition in R, s follows: + 1 k (u v) w + d 1 l ( + 1) + k 1 ( + d 1) + l 1 + ( + k 1) 1 + (d + l 1) 1 + k 1 u d + l 1 u (v w).

3.2. VECTORS AND VECTOR SPACES 47 (P5) Agin, using severl vetor spe properties for (R, +, ) (n you nme them?) we see tht β + 1 β α (β u) α β + 1 β α(β + 1 β) + 1 α α(β + 1 β) + 1 α αβ + 1 αβ αβ + 1 αβ (αβ) u. Therefore, slr multiplition ( ) is n ssoitive opertion. Using distriutive, ssoitive, nd ommuttive lws for (R, +, ), we show tht the distriutive lws hold true for (V,, ).. (P6) Indeed, notie tht + 1 α (u v) α + d 1 α( + 1) + 1 α α( + d 1) + 1 α α + 1 α + α + 1 α 1 αα + 1 α + αd + 1 α 1 α + 1 α α + 1 α α + 1 α αd + 1 α α u α v. Therefore distriutes over. (P7) Notie, lso, tht (α + β) + 1 (α + β) (α + β) u (α + β) + 1 (α + β) α + β + 1 α β α + β + 1 α β α + 1 α + β β + 1 1 α + 1 α + β β + 1 1

48 CHAPTER 3. VECTOR SPACES ( α + 1 α α + 1 α α u β u. ) Thus, distriutes over slr ddition. ( β β + 1 β β + 1 Finlly, we prove the existene of inverses nd identities. (P8) We will find the element z V so tht u z u for every u V. 1 Suppose we let z V. We will show tht z is the dditive 1 identity. Indeed, 1 u z. 1 Thus the dditive identity is, in ft, z. Therefore, property (P8) holds true. (P9) Now, sine u V, we ( will find ) the dditive inverse, ll it ũ so tht 2 u ũ z. Let ũ. We will show tht ũ is the dditive 2 inverse of u. Notie tht 2 + 2 1 1 u ũ. 2 + 2 1 1 Therefore, u ũ z. Thus, every element of V hs n dditive inverse. (P10) Here, we will show tht 1 R is the multiplitive identity. Indeed, 1 1 + 1 1 u 1. 1 1 + 1 Therefore, property (P10) holds for (V,, ). Beuse ll 10 properties from Definition 3.2.1 hold true for (V,, ), we know tht (V,, ) is vetor spe over R. Exmple 3.2.7. Let V R n nd z fixed element of V. For ritrry elements x nd y in V nd ritrry slr α in R, define vetor ddition ( ) nd slr multiplition ( ) s follows: x y x + y z, nd α x α(x z) + z. The set with inry opertions (V,, ) is vetor spe over R (see Exerise 2). )

3.3. THE GEOMETRY OF THE VECTOR SPACE R 3 49 Exmple 3.2.8. Consider the set V R 2. Let ( 1, 2 ) nd ( 1, 2 ) e in V nd α in R. Define vetor ddition nd slr multiplition s follows: ( 1, 2 ) + ( 1, 2 ) ( 1 + 1, 0), nd α ( 1, 2 ) (α 1, α 2 ). Then V with these opertions is not vetor spe (see Exerise 5). 3.3 The Geometry of the Vetor Spe R 3 We n visulize the vetor spe R 3, with the stndrd opertions nd field, in 3D spe. This mens tht we n represent vetor in R 3 with n rrow. For exmple, the vetor v n e represented y the rrow pointing from the origin (the 0 vetor) to the point (,, ) s on the left in Figure 3.2. It n lso e represented s n rrow strting from ny other vetor in R 3 nd pointing towrd point units in the x diretion, units in the y diretion, nd units in the z diretion wy from the strt. (See Figure 3.2 (right).) The nturl question rises: Wht do the v d f g v (d +, f +, g + ) Figure 3.2: Visulizing vetor in R 3 vetor spe properties men in the geometri ontext? In this setion, we will disuss the geometry of some of the vetor spe properties. The rest, we leve for the exerises. Note The geometry we disuss here trnsltes niely to the vetor spe R n (for ny n N) with stndrd opertions nd field.

50 CHAPTER 3. VECTOR SPACES (P1) The Geometry of Closure under Addition To understnd wht it mens geometrilly tht R 3 is losed under ddition, we need to egin y understnding ddition geometrilly. Using the definition of ddition, we know tht if d v nd u f g then v + u + d f + g +. Tht is, v + u is the vetor tht n e represented y n rrow strting t 0 nd pointing towrd the point (d +, f +, g + ). Geometrilly, this is the sme s representing u s n rrow strting t 0 nd the vetor v s n rrow strting t the end of u. We n see in Figure 3.3 tht the sum is the vetor tht strts t 0 (the strt of u) nd points to the end of v. Some desrie this s drwing tip to til euse the tip of u is touhing the til of v. (d +, f +, g + ) v v + u u Figure 3.3: Geometri representtion of vetor ddition in R 3 We see geometrilly tht if we trnslte vetor v long the vetor u, we n form new vetor tht ends t the point to where the tip of v trnslted.

3.3. THE GEOMETRY OF THE VECTOR SPACE R 3 51 (P2) The Geometry of Closure under Slr Multiplition e slr, then α α v α. α Let α Tht is, α v n e represented y n rrow strting t 0 nd ending t the point (α, α, α ). Now, if α > 1, this vetor points in the sme diretion s v, ut is longer. If α < 1, then α v still points in the sme diretion s v, ut is now shorter. Finlly, if α 1, it is the multiplitive identity nd α v v. These n e seen in Figure 5.3. β v v v α v Figure 3.4: Geometri representtion of vetor multiplition in R 3 (0 < α < 1 nd β > 1) Notie tht ny slr multiple of v is represented y n rrow tht points long the line pssing through the rrow representing v. Property (P2) sys tht R 3 ontins the entire line tht psses through the origin nd prllel to v. (P3) The Geometry of the Commuttive Property Notie tht if we trnslte v long the vetor u (see Figure 3.3) or we trnslte u long the vetor v (see Figure 3.5), the vetor formed will point towrd the sme point in R 3. Thus, the ommuttive property shows us tht geometrilly, it doesn t mtter in whih order we trverse the two vetors, we will still end t the sme terminl point. The remining seven vetor spes properties n lso e displyed through similr figures. We leve these remining interprettions in Exerise 6.

52 CHAPTER 3. VECTOR SPACES (d +, f +, g + ) v + u u g v f d Figure 3.5: Geometri representtion of the ommuttive property when ompring to Figure 3.3. 3.4 Properties of Vetor Spes We next present importnt nd interesting properties of vetor spes. These new ides will help guide our further explortions nd provide dditionl tools for distinguishing vetor spes from more generl sets. As you red through these properties, think out wht they men in the ontext of, sy, the vetor spe of imges, or ny of the vetor spes from the previous setion. Theorem 3.4.1. If x, y, nd z re vetors in vetor spe (V, +, ) nd x + z y + z, then x y. Proof. Let x, y, nd z e vetors in vetor spe. Assume x + z y + z. We note tht there exists n dditive inverse of z, ll it z. We will show tht the properties given in Definition 3.2.1 imply tht x y. Indeed, x x + 0 (P8) x + (z + ( z)) (P9) (x + z) + ( z) (P4) (y + z) + ( z) (ssumption) y + (z + ( z)) (P4)

3.4. PROPERTIES OF VECTOR SPACES 53 y + 0 (P9) y. (P8) Eh step in the preeding proof is justified y either the use of our initil ssumption or known property of vetor spe. The theorem lso leds us to the following orollry. Corollry 3.4.1. The zero vetor in vetor spe is unique. Also, every vetor in vetor spe hs unique dditive inverse. Proof. We show tht the zero vetor is unique nd leve the reminder s Exerise 10. We onsider two ritrry zero vetors, 0 nd 0, nd show tht 0 0. We know tht 0+x x 0 +x. By Theorem 3.4.1, we now onlude tht 0 0. Therefore, the zero vetor is unique. The next two theorems stte tht whenever we multiply y zero (either the slr zero or the zero vetor), the result is lwys the zero vetor. Theorem 3.4.2. Let (V, +, ) e vetor spe over F. 0 x 0 for eh vetor x V. Proof. Let (V, +, ) e vetor spe over the field F. Suppose x V. In Exerise 11, we show tht 0 x is the dditive identity in V. Theorem 3.4.3. Let (V, +, ) e vetor spe over R. α 0 0 for eh slr α R. Proof. Assume (V, +, ) is vetor spe over R. Let α R e slr nd let 0 denote the zero vetor. To omplete this proof, we must show tht α 0 is the dditive identity. See exerise 12.

54 CHAPTER 3. VECTOR SPACES Theorem 3.4.4. Let (V, +, ) e vetor spe over F. ( α) x (α x) α ( x) for eh α F nd eh x V. In this theorem it is importnt to note tht indites n dditive inverse, not to e onfused with negtive sign. Over the set of rel (or omplex) numers, the dditive inverse tully is the negtive vlue, while for vetor spes over other fields (inluding Z 2 ) this is not neessrily the se. The theorem sttes the equivlene of three vetors: 1. A vetor x multiplied y the dditive inverse of slr α, ( α) x, 2. The dditive inverse of the produt of slr α nd vetor x, (α x), nd 3. The dditive inverse of vetor x multiplied y slr α, α ( x). While these equivlenes my seem ovious in the ontext of rel numers, we must e reful to verify these properties using only the estlished properties of vetor spes. Proof. Let (V, +, ) e vetor spe over F. Let α F nd x V. We egin y showing ( α) x (α x). Notie tht ( α) x ( α) x + 0 (P8) ( α) x + (α x) + ( (α x)) (P9) ( α + α) x + ( (α x)) (P7) 0 x + ( (α x)) slr ddition 0 + ( (α x)) (Theorem 3.4.2) (α x) (P8) Therefore, ( α) x (α x). Exerise 13 gives us tht (α x) α ( x). Using trnsitivity of equlity, we know tht ( α) x (α x) α ( x). Of prtiulr interest is the speil se of α 1. We see tht ( 1) x x. Tht is, the dditive inverse of ny vetor x is otined y multiplying the dditive inverse of the multiplitive identity slr y the vetor x. 3.5 Exerises Skills Prtie

3.5. EXERCISES 55 1. Let nd e defined on R so tht if, R + + 1 nd + 1. Is (R,, ) vetor spe over R? Justify. 2. Show tht (V,, ) of Exmple 3.2.7 is vetor spe over R. 3. In Exmple 3.2.4, we stted tht T 3 is not vetor spe. List ll properties tht fil to e true. Justify your ssertions. 4. Define nd on R so tht if, R, nd +. Is (R,, ) vetor spe over R? Justify. 5. Show tht the set V of Exmple 3.2.8 is not vetor spe. 6. Drw similr geometri interprettions for the remining seven vetor spe properties not disussed in Setion 3.3. Additionl Exerises 7. Find vetor spe over R, mde of ordered n-tuples (you hoose n), with ddition nd slr multiplition defined omponent wise, whose omponents re in R, ut there re restritions on how the omponents n e hosen? 8. Consider the set R. Define vetor ddition nd slr multiplition so tht vetor spe property (P3) is true, ut property (P6) is flse. 9. Consider the set R. Define vetor ddition nd slr multiplition so tht vetor spe property (P6) is true, ut property (P7) is flse. 10. Prove tht every vetor in vetor spe hs unique dditive inverse to omplete the proof of Corollry 3.4.1. 11. Given tht (V, +, ) is vetor spe over F, show tht if x V then 0 x is the dditive identity in V. Hint: Strtegilly hoose vetor to dd to 0 x nd then use one of the distriutive properties. 12. Complete the proof of Theorem 3.4.3 y proving the following sttement nd then employing Theorem from this hpter. Let (V, +, ) e vetor spe over R nd let 0 denote the dditive identity in V, then α 0 is the dditive identity for V.

56 CHAPTER 3. VECTOR SPACES 13. Complete the proof of Theorem 3.4.4 y writing proof of the needed result. 14. Consider the set of grysle imges on the mp of Afri in Figure 3.1. Crete plusile senrio desriing the mening of pixel intensity. Imge ddition nd slr multiplition should hve resonle interprettion in your senrio. Desrie these interprettions.