The Cloud of Linear Combinations This worksheet is an accompaniment to Section 1.3, Lay's Linear Algebra Written by Michael K. May, S.J., revised by Russell Blyth. Revised by Harry S. Mills Outline: This worksheet is organized by Parts as follows: Part 1 Some "random" discussion. We create a pair of random-number generating functions that we will use to build a random "cloud" of linear combinations from a vector or set of vectors. Part 2 Vectors and Linear Combinations in 2 Dimensions. We see what span means in 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, and what happens when we look at the span of a "pair" of vectors that are not linearly dependent. Part 3 We do the same thing as in Part 2, only in 3 dimensions. We explore the span of three 3-vectors in 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 when they're linearly independent, and what happens when they aren't.

To look at the vector space spanned by a set of vectors it is useful to look at a "cloud" of linear combinations of vectors. We want the "cloud" to consist of essentially random points distributed randomly in 2- or 3-space. restart:with(plots):with(LinearAlgebra): rand0to1 := rand(0..1000)/1000.0: randneg2to2 := rand(-1000..1000)/500.0:

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

We start with some 2-vectors and 3-vectors we can use for computations. 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 v1 := <1, 1>; v2 := <1, 3>;

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

In 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 we can look at a parallelogram of linear combinations bounded by the vectors, by choosing scalars that are between 0 and 1. The original v1 is in red, and the original v2 is in green. The code below generates 100 linear combinations of the form 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. The point here is not to master a lot of code, although you should pick up quite a bit, but to SEE the parallelogram. SpanSetPoints := {seq(rand0to1()*v1+rand0to1()*v2,i=1..100)}: SPoints := pointplot(SpanSetPoints,view=[-5..5,-5..5]): Hv1 := arrow(v1,color=red, shape=harpoon,thickness=3): Hv2 := arrow(v2,color=green, shape=harpoon,thickness=3): display([SPoints, Hv1, Hv2],scaling=constrained);

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

To see a span of our pair of vectors, we allow the coefficients 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 to vary between -2 and 2: SpanSetPoints2 := {seq(randneg2to2()*v1+randneg2to2()*v2,i=1..500)}: SPoints2 := pointplot(SpanSetPoints2,view=[-5..5,-5..5]): display([SPoints2, Hv1, Hv2],scaling=constrained);

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From the above, we're starting to see that the set of all linear combinations of our two 2-vectors will give us the entire plane. For comparison, we can try the same thing with 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 repeated twice: SpanSetPoints := {seq(rand0to1()*v1+rand0to1()*v1,i=1..100)}: SPoints := pointplot(SpanSetPoints,view=[-5..5,-5..5]): Hv1 := arrow(v1,color=red, shape=harpoon,thickness=3): display([SPoints, Hv1],scaling=constrained);

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

The above gives us a line segment in the direction of 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. Now we take our scalars from a set of random numbers from -2 to +2: SpanSetPoints2 := {seq(randneg2to2()*v1+randneg2to2()*v1,i=1..500)}: SPoints2 := pointplot(SpanSetPoints2,view=[-5..5,-5..5]): display([SPoints2, Hv1],scaling=constrained);

6&-I'POINTSG6$%*protectedGI(_syslibG6"6`jl7$$!++++g$)!#5F+7$$"++++)y$!"*F/7$$!+++++k!#6F37$$"++++K5F1F77$$"+++++;F5F:7$$!++++#R"F1F=7$$!++++cIF1F@7$$"++++!G(F-FC7$$!+++++KF1FF7$$!++++CFF1FI7$$"++++/FF1FL7$$!++++w9F1FO7$$"++++WGF1FR7$$!++++)e$F1FU7$$!++++!o"F-FX7$$"++++;RF1Fen7$$"++++/?F1Fhn7$$!++++gPF-F[o7$$"++++_LF1F^o7$$!++++CIF1Fao7$$"++++?&)F-Fdo7$$"++++KLF1Fgo7$$!+++++mF-Fjo7$$!++++g@F-F]p7$$!++++g*)F-F`p7$$!++++/RF1Fcp7$$!++++7>F1Ffp7$$!++++w?F1Fip7$$!+++++SF5F\q7$$"++++G>F1F_q7$$!++++/FF1Fbq7$$"++++![)F-Feq7$$!++++38F1Fhq7$$!++++WQF1F[r7$$!++++kDF1F^r7$$!++++!G"F1Far7$$!++++o<F1Fdr7$$"++++S#*F-Fgr7$$!++++'H#F1Fjr7$$!++++/HF1F]s7$$!++++;OF1F`s7$$!++++%Q#F1Fcs7$$"++++SUF-Ffs7$$"++++'4$F1Fis7$$!++++;IF1F\t7$$!++++C=F1F_t7$$"++++S%)F-Fbt7$$!++++3?F1Fet7$$!++++CHF1Fht7$$!++++'4#F1F[u7$$"++++cOF1F^u7$$!++++![$F1Fau7$$"++++wCF1Fdu7$$"++++wGF1Fgu7$$"++++sDF1Fju7$$!++++;8F1F]v7$$"++++K=F1F`v7$$!++++)y$F1Fcv7$$"++++!3$F1Ffv7$$!++++W8F1Fiv7$$!++++?NF-F\wF.7$$"+++++!)!#7F_w7$$"++++%)GF1Fcw7$$"++++7MF1Ffw7$$"++++%o#F1Fiw7$$"++++gJF1F\x7$$!++++)3#F1F_x7$$"++++!o'F-Fbx7$$!++++WJF1Fex7$$!++++!3$F1Fhx7$$!++++c6F1F[y7$$!++++wQF1F^y7$$"++++;MF1Fay7$$!++++/BF1Fdy7$$!++++s<F1Fgy7$$!++++!y#F1Fjy7$$"++++/AF1F]z7$$!++++'f$F1F`z7$$!++++w@F1Fcz7$$!++++!3#F1Ffz7$$"++++c5F1Fiz7$$!++++SPF1F\[l7$$!++++39F1F_[l7$$"++++!o&F-Fb[l7$$"++++)G"F1Fe[l7$$"++++!3%F-Fh[l7$$"++++s=F1F[\l7$$!+++++wF5F^\l7$$!++++'H$F1Fa\l7$$!++++g6F1Fd\l7$$"++++gHF-Fg\l7$$"++++gRF1Fj\l7$$!++++OLF1F]]l7$$!++++O<F1F`]l7$$!+++++MF1Fc]l7$$"++++S?F1Ff]l7$$"++++GNF1Fi]l7$$!++++_KF1F\^l7$$!++++SsF-F_^l7$$"+++++kF5Fb^l7$$"+++++#*F5Fe^l7$$!++++?@F-Fh^l7$$"++++o7F1F[_l7$$!++++)e"F1F^_l7$$!++++s7F1Fa_l7$$"++++?VF-Fd_l7$$!++++?JF1Fg_l7$$!++++W?F1Fj_l7$$!++++C;F1F]`l7$$!++++kAF1F``l7$$!++++gZF-Fc`l7$$"+++++sF5Ff`l7$$"++++![%F-Fi`l7$$"++++C?F1F\al7$$"++++?EF1F_al7$$"++++wNF1Fbal7$$!++++[OF1Feal7$$"++++S[F-Fhal7$$!++++))HF1F[bl7$$"++++oOF1F^bl7$$"++++?rF-FablFa[l7$$"++++![$F-Fdbl7$$"++++)o$F1Fgbl7$$!++++_OF1Fjbl7$$!++++k;F1F]cl7$$"++++'p"F1F`cl7$$!++++KEF1Fccl7$$"++++38F1Ffcl7$$"+++++SF5Ficl7$$"++++37F1F\dl7$$!++++SQF1F_dl7$$!++++S[F-Fbdl7$$!++++#p$F1Fedl7$$!+++++IF-Fhdl7$$!++++CPF1F[el7$$!++++kFF1F^el7$$"++++)[#F1Fael7$$!++++?>F1Fdel7$$!++++3OF1Fgel7$$!++++!["F-Fjel7$$"++++!))*F-F]flF^^l7$$"""""!F`fl7$$!++++SIF-Fdfl7$$!+++++))F-Fgfl7$$!++++WNF1Fjfl7$$!++++'\#F1F]gl7$$F]wF1F`gl7$$!++++WKF1Fbgl7$$FeelF-Fegl7$$"+++++%*F-Fggl7$$!++++[:F1Fjgl7$$!++++3@F1F]hl7$$!++++KMF1F`hl7$$!++++[8F1Fchl7$$"++++gNF1Ffhl7$$"+++++UF-Fihl7$$"+++++gF5F\il7$$"++++SHF1F_il7$$"++++WEF1Fbil7$$"++++wMF1Feil7$$!++++CDF1Fhil7$$"++++gTF-F[jl7$$!++++7;F1F^jl7$$!++++k>F1Fajl7$$!++++3LF1Fdjl7$$!++++oBF1Fgjl7$$"++++/JF1Fjjl7$$!"%FbflF][m7$$"++++#p$F1F`[m7$$!++++?DF1Fc[m7$$"++++?TF-Ff[mFep7$$"++++WBF1Fi[m7$$!++++gMF1F\\m7$$!++++K9F1F_\mFE7$$"++++/IF1Fb\m7$$"++++?9F1Fe\m7$$!++++g7F1Fh\m7$$"++++%y#F1F[]m7$$!++++g;F1F^]m7$$"++++k=F1Fa]m7$$!++++OJF1Fd]m7$$!++++cPF1Fg]m7$$"++++g`F-Fj]m7$$"++++OOF1F]^m7$$"++++kPF1F`^m7$$!+++++sF5Fc^m7$$!++++_;F1Ff^m7$$F4F-Fi^m7$$!++++!3(F-F[_m7$$!++++?HF-F^_m7$$"++++OFF1Fa_m7$$"++++;OF1Fd_m7$$"++++oLF1Fg_m7$$!++++!o&F-Fj_m7$$!++++SMF1F]`m7$$"++++!)QF1F``m7$$!++++GIF1Fc`m7$$!++++S!*F-Ff`m7$$!++++#z#F1Fi`m7$$"++++?bF-F\amFes7$$!++++3KF1F_am7$$F]qF-Fbam7$$!++++s6F1Fdam7$$"++++)=$F1Fgam7$$!+++++QF-Fjam7$$!++++w5F1F]bm7$$"+++++WF5F`bm7$$!++++cNF1Fcbm7$$FbflFbflFfbm7$$"++++C=F1Fhbm7$$"++++o?F1F[cm7$$"++++O?F1F^cm7$$"++++GQF1Facm7$$FghlF-FdcmF\^m7$$!++++?BF1Ffcm7$$!+++++GF5Ficm7$$!++++S5F-F\dm7$$!++++cAF1F_dm7$$"++++#R#F1Fbdm7$$!++++%o$F1Fedm7$$"++++!)HF1Fhdm7$$!+++++6F1F[emF67$$!+++++OF1F^em7$$!++++O7F1Faem7$$!++++![%F-Fdem7$$!++++3=F1Fgem7$$"++++gjF-Fjem7$$"++++W9F1F]fm7$$!++++7<F1F`fm7$$!++++K7F1Fcfm7$$!++++oCF1Fffm7$$!++++3GF1Fifm7$$!++++!3"F-F\gm7$$"+++++[F-F_gm7$$!++++kBF1Fbgm7$$"++++kAF1Fegm7$$!++++S;F1Fhgm7$$"++++3KF1F[hm7$$Fc^lF-F^hm7$$!++++!)HF1F`hm7$$"++++35F1FchmFigl7$$!+++++7F5Ffhm7$$"++++CIF1Fihm7$$"++++GGF1F\im7$$!++++gLF-F_im7$$"++++oJF1Fbim7$$!++++_?F1Feim7$$!++++c8F1FhimF[q7$$"++++%G#F1F[jm7$$"++++GFF1F^jm7$$"++++;HF1Fajm7$$"++++gFF-Fdjm7$$"++++'\$F1Fgjm7$$"++++'>"F1Fjjm7$$!++++_JF1F][n7$$"++++s7F1F`[n7$$"++++GMF1Fc[nFh]l7$$"++++3=F1Ff[n7$$!++++!)GF-Fi[n7$$"++++O7F1F\\n7$$!++++/<F1F_\n7$$!++++%e"F1Fb\n7$$!++++S]F-Fe\n7$$"++++wOF1Fh\n7$$"++++;AF1F[]n7$$!++++'*=F1F^]n7$$!++++GRF1Fa]n7$$!++++[NF1Fd]n7$$!++++SEF-Fg]n7$$!++++3CF1Fj]n7$$"++++W8F1F]^n7$$!++++C9F1F`^nFa\mFadlFefm7$$!++++o5F1Fc^n7$$"++++SYF-Ff^nF_v7$$"++++cFF1Fi^n7$$"++++KFF1F\_n7$$"++++'*HF1F__nFcel7$$"++++/KF1Fb_n7$$"++++_KF1Fe_n7$$"++++oIF1Fh_n7$$!++++o?F1F[`n7$$!++++!=#F1F^`nF`y7$$"++++SuF-Fa`n7$$!++++_9F1Fd`n7$$"++++!["F-Fg`n7$$!++++'*GF1Fj`n7$$"++++O6F1F]an7$$"++++S!*F-F`an7$$"++++)Q$F1Fcan7$$!++++KPF1Ffan7$$!++++w;F1Fian7$$!++++O:F1F\bn7$$"++++C@F1F_bn7$$"++++?lF-Fbbn7$$"++++k<F1Febn7$$"++++s5F1Fhbn7$$FjcmF1F[cn7$$!++++kEF1F]cn7$$"++++?8F-F`cn7$$!++++o6F1Fccn7$$!++++[IF1Ffcn7$$"+++++eF-FicnF]]m7$$"++++WDF1F\dn7$$!++++SmF-F_dn7$$!++++gzF-FbdnF67$$!++++/8F1Fedn7$$"++++/@F1Fhdn7$$"++++sRF1F[en7$$"++++!G#F1F^en7$$"++++w7F1Faen7$$!++++s;F1Fden7$$!++++!)=F1Fgen7$$"++++kFF1FjenFevFc_m7$$"++++[BF1F]fn7$$"++++?=F1F`fn7$$"++++C6F1Fcfn7$$!++++SAF-Fffn7$$!+++++iF-Fifn7$$!++++oIF1F\gn7$$!+++++=F1F_gn7$$"++++w8F1Fbgn7$$"++++oCF1Fegn7$$!++++kOF1FhgnF[w7$$!++++g8F-F[hnFj\m7$$!++++?$)F-F^hnFhcn7$$"++++[7F1Fahn7$$!++++O9F1Fdhn7$$"++++'*QF1Fghn7$$"++++'4"F1Fjhn7$$!++++G8F1F]in7$$"++++ORF1F`in7$$!++++/5F1Fcin7$$!++++?;F1Ffin7$$!++++KOF1Fiin7$$"++++OGF1F\jn7$$"++++SOF1F_jn7$$"++++77F1Fbjn7$$"++++gBF-Fejn7$$"++++CMF1Fhjn7$$!++++#f#F1F[[o7$$!++++OGF1F^[o7$$"++++'>#F1Fa[o7$$"+++++PF1Fd[oFgn7$$FabmF-Fg[o7$$"++++c>F1Fi[o7$$!++++/@F1F\\o7$$!++++;5F1F_\oFew7$$"++++?^F-Fb\o7$$!++++C<F1Fe\o7$$"++++O<F1Fh\oFb^mFjn7$$!++++?fF-F[]o7$$"++++?\F-F^]o7$$"++++o>F1Fa]oFh]l7$$F\hnF1Fd]o7$$"++++7NF1Ff]o7$$!++++K?F1Fi]o7$$!++++KBF1F\^o7$$!+++++cF-F_^o7$$!++++;LF1Fb^oF[[l7$$"++++g8F1Fe^o7$$"++++)e#F1Fh^o7$$!++++wGF1F[_o7$$"++++g;F1F^_oFfjm7$$"++++g$*F-Fa_o7$$"++++;EF1Fd_o7$$!++++?:F1Fg_o7$$"++++))RF1Fj_o7$$"++++!3#F1F]`oFc_o7$$"++++kLF1F``o7$$!++++3BF1Fc`o7$$!++++)Q$F1Ff`o7$$"++++[RF1Fi`o7$$"++++SFF1F\ao7$$!++++#\"F1F_ao7$$!++++#4"F1Fbao7$$F`wF5Feao7$$"++++WPF1Fgao7$$!++++3<F1Fjao7$$!++++OBF1F]bo7$$!++++gKF1F`bo7$$!++++'*>F1Fcbo7$$"++++7OF1Ffbo7$$!++++CMF1Fibo7$$!++++?vF-F\co7$$"++++WOF1F_co7$$!++++gHF-Fbco7$$!++++S'*F-Feco7$$"++++SgF-FhcoFcu7$$"++++oMF1F[doFd[l7$$!++++3MF1F^do7$$!++++?6F-Fado7$$"+++++9F1FddoF]do7$$"++++S=F1Fgdo7$$!++++#p#F1Fjdo7$$"++++gdF-F]eo7$$"+++++uF-F`eo7$$"++++o@F1Fceo7$$!++++3NF1Ffeo7$$!++++o;F1Fieo7$$"++++s<F1F\fo7$$!++++_>F1F_foFa\m7$$"++++K;F1Fbfo7$$!+++++;F1Fefo7$$!++++%)QF1Fhfo7$$"+++++:F1F[goFcuF^gm7$$!++++o=F1F^go7$$"++++wKF1Fago7$$Fh\lF1Fdgo7$$!++++g$*F-Ffgo7$$"++++!)oF-Figo7$$"+++++%)F5F\hoF[xF^dn7$$!++++S<F1F_hoFjtF27$$!+++++')F-Fbho7$$"++++?PF-Feho7$$!+++++'*F-Fhho7$$!++++SoF-F[io7$$!++++S@F1F^io7$$"++++/PF1Faio7$$"++++[HF1Fdio7$$"++++w=F1Fgio7$$"++++SBF1Fjio7$$"++++![&F-F]jo7$$"++++/LF1F`jo7$$!++++gTF-Fcjo7$$"++++ScF-FfjoFjho7$$!++++!G$F1Fijo7$$!++++?<F1F\[p7$$!++++?7F1F_[p7$$!++++'4$F1Fb[p7$$"++++gDF-Fe[p7$$"++++;IF1Fh[pFc]nFcamFg\mFhw7$$"++++?xF-F[\p7$$!++++gvF-F^\p7$$F`dlF-Fa\p7$$!++++?OF1Fc\p7$$"++++sKF1Ff\p7$$FbuF-Fi\p7$$FGF-F[]pF^[l7$$!++++sRF1F]]p7$$"++++_>F1F`]p7$$!++++g<F-Fc]p7$$"++++[?F1Ff]p7$$!++++sFF1Fi]pF[co7$$"++++KAF1F\^p7$$!++++sPF1F_^p7$$!++++))RF1Fb^pFdimF`jm7$$!++++s5F1Fe^p7$$!+++++[F5Fh^pFggm-I'CURVESGF%6&7%FebmF_fl7$$"+++++vF-$"+++++&)F--I&STYLEGF%6#I,PATCHNOGRIDGF%-I'COLOURGF%6&I$RGBGF%$"*++++"!")FfbmFfbm-I*THICKNESSGF%6#""$-I(SCALINGGF(6#I,CONSTRAINEDGF%-I%VIEWGF%6$;$!"&Fbfl$""&FbflFi`p

Now we try the same thing in 3D: Define three 3-vectors, 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, 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, and 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: w1 := <1,1,1>; w2 := <-1,3,2>; w3 := <2,3,-1>;

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Now we plot a random list of linear combinations of the 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's, taking our scalars from the interval [0,1], as before: SpanSetPoints := {seq(rand0to1()*w1+rand0to1()*w2+rand0to1()*w3,i=1..500)}: SPoints := pointplot3d(SpanSetPoints,view=[-5..5,-5..5,-5..5]): Hw1 := arrow(w1,color=red, shape=harpoon,thickness=3): Hw2 := arrow(w2,color=green, shape=harpoon,thickness=3): Hw3 := arrow(w3,color=blue, shape=harpoon,thickness=3): display3d([SPoints, Hw1, Hw2, Hw3],scaling=constrained, axes=boxed);

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By grabbing and rotating the image above, you can see how it has some real "volume" to it. The idea is that these three 3-vectors span all of 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. These three 3-vectors are linearly independent. To see this better, we can let the scalars run from -2 to + 2 as before, and see what kind of "cloud" is produced: SpanSetPoints2:={seq(randneg2to2()*w1+randneg2to2()*w2+randneg2to2()*w3,i=1..500)}: SPoints2 := pointplot3d(SpanSetPoints2,view=[-5..5,-5..5,-5..5]): display3d([SPoints2, Hw1, Hw2, Hw3],scaling=constrained, axes=boxed);

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One should postulate that the set of all linear combinations of the three 3-vectors will, in fact, fill all of 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. As before, interesting things happen when we repeat vectors and use a trivially non-independent set, and examine all linear combinations of the form 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. This, of course, is equivalent to linear combinations of 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 and 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. Just a pair of 3-vectors in 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. What do you suppose their span will be? Pay careful attention to the definition of Hw3, in the code below. It is not the same "Harpoon Vector 3" that we saw in the preceding section. SpanSetPoints := {seq(rand0to1()*w1+rand0to1()*w2+ rand0to1()*(w1+w2),i=1..500)}: SPoints := pointplot3d(SpanSetPoints,view=[-5..5,-5..5,-5..5]): Hw1 := arrow(w1,color=red, shape=harpoon,thickness=5): Hw2 := arrow(w2,color=green, shape=harpoon,thickness=5): Hw3 := arrow(w1+w2,color=blue, shape=harpoon,thickness=5): display3d([SPoints, Hw1, Hw2, Hw3],scaling=constrained, axes=boxed);

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This suggests that the span of these three 3-vectors is only two-dimensional, as does the following Maple: SpanSetPoints2 := {seq(randneg2to2()*w1 + randneg2to2()*w2+randneg2to2()*(w1+w2), i=1..500)}: SPoints2 := pointplot3d(SpanSetPoints2,view=[-5..5,-5..5,-5..5]): display3d([SPoints2, Hw1, Hw2, Hw3],scaling=constrained, axes=boxed);

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