Vector space properties MA 751 Part 1 Linear Algebra, functional analysis This material is from basic linear algebra as a reference - will not go over in detail in class. 1. Preliminaries: Recall: ‘‘ = {real numbers}; $ = {all triples of real numbers}, etc. Examples Definition (part 1): A vector space is a set of objects Z on which addition and scalar multiplication are defined. Precisely ifvv"# ,−Z we define vv"# and if vv−Z we define - for all -−‘ . Elements of Z are called vectors. [Additional required properties are below]. Examples Ex 1: ‘$: Examples Geometric representation of vector v: v œ @ ß@ ß@ œ arrow from origin to Ð@ @ ß@ ÑÀ ¡"#$ "ß#$ Examples Addition: "ß #ß $ #ß "ß " œ $ß $ß # ¡ ¡ ¡ Scalar multiplication: # "ß #ß $ œ #ß %ß ' ¡ ¡ Vector space properties Properties: 1. uvvuœ 2. uvwРќРuvw Ñ etc - Vector space properties 3. There exists a unique vector ! Ð œ !ß !ß ! Ñ such that ¡ vv!œ for all vectors v. 4. For each vv−Z, there exists a unique element −Z such that vv œ! 5. for uvß−Z and -−-ÐÑœ--‘ u v u v 6. for -ß . −‘ and uuuu − Z, Ð- .Ñ œ - . 7. -Ð.uu Ñ œ Ð-.Ñ for -ß .ß u as above 8. for uuu−Z, "† œ Remark on 4: clear if vvœÐ@ßáß@Ñ"8 then œÐ@ßáß@Ñ " 8 in special case Zœ‘8 Þ Vector space properties Def. (part II - full definition): A vector space is a set of objects Z on which addition and scalar multiplication are defined with properties 1 - 8 above. Examples % Ex. 2: ‘ , e.g. ØB"#$% ßB ßB ßB Ù viewed geometrically (arrows) or in terms of vector algebra. Ex 3: Zœ all polynomials of degree Ÿ# form a vector space: # v œ+!" +B+B # Examples For example we have for ## vv"#œ$#B%Bà œ#B$B # vv"#œ&BB # $v" œ * 'B "#B . Properties 1 - 8 apply, e.g.: vvvv"#œ #" %Ðvv"# Ñ œ % v " % v # Þ Examples Ex 4: Zœ continuous functions 0ÐBÑ on Ò!ß1‘ Ó § # @œ"#sin Bà @œBÎ"! Examples 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 Examples Add vectors: # vv"# œsin B B Î"! Scalar multiply: $œ$Bv" sin Check vector properties (1-8) satisfied. [vector space can be: arrows, polynomials, continuous functions; need only define addition, scalar mult, and vector properties (1-8) must hold]. Properties 2. Vector space properties: Def 1: Given vector spaceZßßá−Zß and vv"# v 8 a vector wvv is a linear combination (LC) of "8, á , if wvœ-"" á - 88 v for some collection Ö-"8 ßáß- ק ‘. Properties Ex 5: vœ "ß#ß$ à v œ #ß"ß" à v œ "ß#ß" à "# ¡ ¡ $ ¡ w œ &ß )ß ( ¡ is linear combination of v3"#$À-œ"à-œ#à-œ#can let above. Properties #BB Ex 6: vvv"#$œB#BàœB/àœ/sin sin #B wvœBB#/3sin 6 is LC of the 3, since wvvvœ$"#$ ! # Def 2: Given vector space ZÖßßáקZ the collection vv"# v 8 is linearly independent (LI) if no v3 is a LC of the others. Properties Note: if b-ßá-! constants"8 (not all ) s.t. -á-œ!ß""vv 88 then say if -Á!" , @œ-Î-@á-Î-@ß"#""8"8 so one vector is a LC of others and so not lin. ind. Can reverse to show: Theorem: v"8á v are lin. ind. iff do not exist const. - " ßáß- 8 (some non-zero) s.t. c""vvá - 88 œ!Þ Properties Ex 7: vœ "ß#ß$ à v œ #ß"ß" à v œ "ß#ß" "# ¡ ¡ $ ¡ are LI Reason: if ---œ!ß""v ## v $$ v - "ß #ß $ - #ß "ß " - "ß #ß " œ !ß !ß ! "# ¡ ¡ $ ¡ ¡ Ê-#--œ!"#$ #-"# - #- $ œ! $-"#$ - - œ! Êsolving, -"#$ œ- œ- œ!. Properties #BB Ex 8: vvv"#$œsin B#Bà œ# sin B/ à œ/ are LI Proof: -œ!33v holds only with - 3 all 0; plug in several B 3 values to show thisß e.g., set Bœ!ßßÎ#11 in the equation -œ!33v to obtain 3 separate equations from which it 3 follows all -œ!3 . [note also that one of the vectors would have to be a combination of the other two for all B; plausible that this cannot happen] Basis and dimesion 3. Dimension of a vector space: Given a vector space Z Def 3: We say that the set of vectors Wœ Ö@ßá@×"8 span Z if every vector A−Z can be written as a linear combination of vectors in W. Basis and dimesion Ex 9: @ œ "ß #ß $ à @ œ #ß "ß " à @ œ "ß #ß " span "# ¡ ¡ $ ¡ Zœ‘$ Þ To see this, note that we have to show that for any vector Aœ AßAßA ß we can find constants - ¡"#$ 3 such that -@"" -@ ## -@ $$ œAÞ [Again involves solving a system of equations; see exercises] Basis and dimesion Def. 4: The set of vectors WœÖvv"8 ßáß ×§Z is a basis for Z if (i) WZ spans (i.e., any vector A−Z can be written as a linear combination of vectors from WÑ, and (ii) the vectors from W are linearly independent. Theorem 1: A set W§Z is a basis for Z iff every w −Z can be written uniquely as a linear combination of vectors in W. Ex 10: vvœ "ß#ß$ à œ #ß"ß" à v œ "ß#ß" are "# ¡ ¡ $ ¡ a basis for ‘$Þ Basis and dimesion 3 Every vv− ‘ can be written as a LC of the 3 uniquely. For example w´ $ß (ß ' œ " † vvv " † # † , ¡"#$ is only LC of the vw3 giving . Basis and dimesion #BB Ex 11: vv"#œsin B#Bà œ# sin B/ à v $ œ/ are a basis for the vector space Zœ all linear combinations of sin BßB#Band /. #B œÖ-"#$3sin B- B -/ À - −‘ ×. Note: All bases for vector space Z. have same number of elements. .œ dimension of Z. Vector norms 3. Norm (length) of a vector: Notation: In ‘$ write @" v œØ@ß@ß@Ùœ"#$ Ô×@# ÕØ@$ Def. 5: Norm vvœ² ²distance from end to origin 222 = È|@@@"#$ | + || +| | Vector norms Vector norms 4. Distance between vectors $" Between uvœœÔ×## and Ô ×: ÕØ"" Õ Ø #### .œÈ Ð$"Ñ Ð#Ð#ÑÑ Ð""Ñ œmuv m Vector norms Geometry:.