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planerotplanerotanswerstex matgenmatgenanswerstex compmapcompmapanswerstex inversesinversesanswerstex modmmodmanswerstex eigeneigenanswerstex compfunccompfuncanswerstex geometry of complex numbers problemstart explain why is perpendicular to without using DeMoivre's theorem let then which the transformation drawing this out on D plane makes clear that angle between two points and origin simply by subtracting angles shown in Figure width pair a O Ap Af Apf draw Arrow dotA dotO dotAp NE NW drawOA drawOAp drawAAf dashed drawApApf ApApf N OApf E AfA OAf S path arc arcO expiatan drawarc as long you can also rationalize it fact lines through have slopes respectively so they must be problem how does relate Hint symmetry again flipped over axis since imaginary part negated thus due congruent triangles formed if we are pedantic we'd either use multivalued function or say modulo geometric interpretation real markscalefactor picture pathtickspath G int r spacing pen pcurrentpen pict B C direct T larclengthg spacespacingmarkscalefactor halfticksmarkscalefactor startpt directunitdirg arctimeg rl startptrlnspace for i tstartptispace Bpointg halftickdirect CB drawpict BC p return z zconj SE zf dotzf drawOz drawOzconj drawzzconj addpathticksOz [number] addpathticksOzconj addpathtickszfz addpathtickszfzconj addpathticksOzf arcc drawarcc arrows pointarcc form showing compute once more other words square distance from basically just an application sohcahtoa triangle details divide rationalizing denominator abicdicdicdi abicdicd acbdbcadicd don't rule yet applying division but quickly derive rr multiplying conjugate compare contrast methods problems convenient depend circumstance opinions may vary definitely faster do dividend divisor already likely than converting rectangular back what hinted at previous shows having abiab pencil paper brains only no calculators forms with calculator convert apply ii iii your prove expression add them vectors W sqrt drawOw drawwz dotw dotz dir pointOw wz drawwsqrt figureAddition first vector starting second endpoint along isosceles furthermore apex has measure radians base adding our desired analyze wish find tangent know zz solve show its solutions under operation multiplication group isomorphic rotation equilateral write table there's numerous ways let's usual k kZ these three values indeed particular identity counterclockwise following groups nonsquare rectangle choose regular hexagon footsteps equivalent elements should obvious rotations make each defined natural way raising power hexagon's dihedral consider same different difference fundamental not although size periods while share some subgroups however trivial subgroup generated sets addition closed 's inverse reasons though all properties satisfied can't such satisfies associativity associative anyways invertibility element their own closure sum any takes set one easy see multiples because cannot satisfy there example reason integers another integer except multiplicative inverses rational number longer needs now product rationals brute force anglesum trig identities prefer method page gives better understanding works actually it's kind satisfying underlying mechanics probably done most work where equating parts friend's textbook says who's right both remembering pretty easily change finish rest proof need context answer check relevant section approximately quadrilateral squareOnpair ABBrotateABArotateBAcycle BA DC ad rotatea q rotateb rotatec SS rotated drawABCDcycle SW NNW NEE drawsquareOnA drawsquareOnB drawsquareOnC drawsquareOnD dotB dotC dotD dotP dotQ dotR dotSS drawPR drawQSS figureThe relative magnitudes directions below halving sides m Rp QSS drawOb drawOc drawOd drawOm drawOn dotm dotn segments length justification get midpoint go up center process ib drawAB AB drawBBb BBb drawBbBbib BbBbib formulae terms similar fashion note going interpretations ic id drawAAid AAid drawAidSS AidSS drawAD drawDDc DDc drawDcDcic DcDcic evaluate straightforwardly abibdididcicaia ia abcdiaibicid abcdi drew squares outside connected centers conjecture happens connect construction leads connecting together variables we'll labeled ac triangleOnpair v rotate cycle gray drawABCcycle drawtriangleOnA drawtriangleOnB drawtriangleOnC drawBQBbcycle dotted drawBb Bbib drawPQRcycle figureEquilateral around central arbitrary last loss generality define halfway finally vertex outlined line combining facts logic like after magnitude SAS Congruence substituting found iaic abibia ibia ibiai iaibiaib iaib ABC tedious worked drawO PR dotPR QP dotQP pointarc transform sx shift addpathticksOPR addpathticksOQP drawsx Pp drawPpQpRpcycle dotPp dotRp addpathticksPpQp addpathticksPpRp arcP pointPR midpointarc figureSAS lets us sufficient hard try sure easier starts setting diagram algebraically ang pi tanang f drawOfz fz expiang Express matches expand factor into octagon corresponding pentagon tables flip about yes even order easiest latter former defines follow commutative quickest noting isomorphism cyclic name subsets infinite finite property list few type generalize simple examples general th roots unity gon interpret expressions implies away translated unit quotient scalar times argument equal Scenario expi expiatanzy zx Oz drawzz arcz unitz figureA graph parallel Postulate marked double truly complete considered quadrant case extending locations nonetheless provide algebraic solution fun supposed interesting scale polar equation hinting traced cardioid produced import freal cost gpolargraphf operator xaxis yaxis abovetrue drawg dotLabel guys geometrically various mean means reflection across units halve component height new point quantity dirang omz dotomz zomz drawOzomzcycle drawOfzz nv nvf nvx drawOnvnvfO dotnv expiatannvy labelrotateang Onv drawshiftscale unitOomzbraceO unitnvnvfbracenv ras drawshiftnvf rasras figureGraph lengths indicated braces couple perhaps nvm drawOnvm Onvm nvmnv nvmnvf drawnvmnvnvfcycle dotnvm dotnvf succulent hypotenuse equidistant vertices pairing previously technically slightly incomplete exist described extend cases I'd give perks halfangle abi bi personally strongly quick understand idea true suggests expanding would rough take exponents inside symbolsThis dumb many polynomial degree algebra aren't repeated appear factorization twice unfortunately think calculusFor route isn't pleasant until clever rearrangement substitution start reveals symmetric zy arcr drawarcO pointOunitz atanzy arcrexpiang arcrexpipi residents we're mathematical convenience fourth quadrantsIf you'd taken clockwise positive unique restrict range wait issue premise strict relation intersect solving agreement claim ei eii definitions eiei was expected eiiei eiieiei multiple additive correspond you've used quadratic throughout High School cubic finds combine create ybbybcybd ybybyb bybyb cybc y bby cy bbbcd cby cpy dq rearrange substitute simplify st pst ps pt sst tst sp tp checks eliminate equations ststsqs qs above inverted sign when negative vice versa formula taking cube sq tq keep mind rooting possible pk root principal value multiply preserve otherwise produce result pq could monstrous thanks started coefficient front term well come attack did remember proved obtain proceed pyq Cp pcp provided yield namely adds yields observed strategy still larger domain certainly steps were doozy cgproblemi distancepair sqrtpx px py markanglelabelprecisepair string labelc direction mindistancep distancep arca pointpp labellabelc midpointarca unitifypair dis sqrtcx cx anglemarkingpair labeldis markanglelabelprecisep unitifyp rightanglepair pc rangle Tristan Needham's Visual analysis problemsexamples Josh Zucker text dot iz dotiz drawanglemarkinga drawrightanglez Az doti drawanglemarkingz drawanglemarking zbrace bracez izbrace braceAz bracestyle blackbp drawzbrace drawizbrace midpointzbrace midpointizbrace Breaking components observe year became masters art manipulating will build background book reinforce skills Dose Vitamin least DeMoivres recall making rewrite notice multiplied addedWhat else added thats story repeatedly original reduces periodicity coterminal therefore duplicates confirm factoring pure being depicts carried observation knowledge scaling effect rotating proving notation often called written bar top universal sense calculate every suppose want cis definition cs Binomial expansion rotatepair degrees rads cosrads sinrads pxcpys pycpxs ABBrotateAB ArotateBA AA Bb Cc Dd drawAAa drawAaB drawBbC drawCCc drawCcD drawDDd drawDdA four drawAa drawaP arotatea construct joining opposite represent edges thought Subtraction edge similarly etc course cgproblemii composition functions depends appears period always unless undefined look results iterating initial record noticed fundamentally types behavior iteration depending theories expect given identify behaves differently compositions itself generate big iterate illustrates plug time pattern looking onetoone correspondence compared matrices computer systemif you're lazyto my writing answers nonzero fixed periodic plot ranges circle rays associated prediction predict hypothesis remains locate had variety completing doing able characterize remain help successive reflections intersecting intersection chose close happen thrice test picking move farther produces rotationlike competing nearer exerting influence distorted compose feel free system inner left outer repeat decompose involving conjugates describe involve recompose visualize reflect invert processdon't end keeping track ones inversion play role attractive repelling nonreal sequential eigenvalues reminds apparent discrepancy chapter determinants ever themselves eigenvalues' investigation composing numerical experimentation we've investigated transformations turn composed decomposed behave reviewed themes acquainted review begin rather efficiently enter next hit key clearly approach fxxxxxx fxxxxxxx tabulate reached temporarily narrow focus class affects fxDxx dxx constant later fxfffxffDx fDDx fDxDDx DDxDDx x ddxdxdxdxx dxdxdxx DDxx recognize nasty omitted fxfffxffDDxDDx DDxDDxDDxDDxDxx DxxDD getting annoying computing comparing job corresponds DDD fx DxDDx FF DDDD DDDDD DDxDxDD six things five repeats iterations necessarily question attention discovered looks home rename dealing continue good select approaches neither reasoning behind unclear questions investigate view been decomposing shed light subject willl obscure useful preimage very image almost identical circle's circumference depicted figures transformpair pairlist labelsP pairc pairlisti pairtr transformpairc drawpaircpairtr dotpairtr dotpairc pxpxpypy dirsNW dirsi dots cospi sinpi conjugation causes oscillate insights correct acts usually changes independent eigenvectors act composite mappings fill blank finding images that's those intermediate coordinate net represents achieve mapping rightmost didn't leftmost hand familiar reverse Bp AAp BBp dotBp drawpointAc pointAc drawpointBc pointBc followed overall ordering working aka moved half wanted read wrote representing row transpose here format convention evaluated sequentially Snaps Flips transportation destinations columns origins side rows unlike classes comprise isometries really planar operate directed generalized boring comes matrix's substituted confusing gotten oh specific map centered stretching shearing ideas quantitative specifying andor factorproblemstart nothing scales dilates rotates initially mysterious realize rewritten stretch squish shear seems bit foreign meaning letting parametrizes standard sane person's negation earlier column precisely parameterized coordinates unfortunate hope matrixtransformation undoes instance undone doesn't mapped effects multiplications outcome void graphMatrixreal unitab cd unitba graphMatrix dilation matter linear combination necessary products minimum others justify choice stretches shears seem include possibilities benefit foresight perform applied follows graphMatrixsqrt reversed projection onto might old foot altitude total impossible undo project parallelogram area information BD drawB ENE pointAD pointAB abrace braceD drawabrace rotateshift anglemark rotatepointAB rotatepointAD drawBrotate drawanglemark rotateAB drawrotate ESE patterns addbrick brick L drawAKKKcycle drawLLCLcycle filldrawAKBcycle patternbrick filldrawAKDcycle filldrawLBKDcycle filldrawBLCcycle filldrawDCLcycle put contains maps diagonals rises descends subtract entries ascending diagonal descending formulas relationship determinant measures absolute differ conserved orientation handedness chirality whatever call preserved indicate indicates amount areas scaled changed zero degenerate figureScenario rectangles dimensions sentence explaining seven minus shaded bricks carry large wide tall small switch places calculated isometry included yielding require reversal imply transformational reverses happened replaced become kC BkD kD drawBkCkDA dotkD dotkC gets constructing collinear collapses drawForreal sy Dp maxx maxa apcp minx mina maxy maxb bpdp miny minb drawminx bac bdc dcycle bpapcp bpdpcp dpcycle labelNot drawFor iv vi undoing visualized step operations basic important auto overlay pic csa bend node csb csc csd filled decomposition explained third final probneededformatrixundoHere ex reflects cont sequence backwards matricesproblemstart lacks projecting laterally whole everything pgray action nullified simplicity full aside here's arises bottomright corner projects stuck unknowns expands choosing giving bruhcount far identified including seen interact egg j filltypeFillDrawfillpenwhite drawpenblack filltype fillc FillDrawfillpenblack currentpen rot drawsubpathrot aware speaking reduce dimensional intuitive dilations repertoire retrace arrived recipe improve outline firstpara checking existWhy becomes shrink topleft entry bcabd cabcad caadbca bottom reiterate filling forget ultimate goal fortunately em known simpler pieces sample cover cisreal theta costheta sintheta cispi corners Draws drawA current drawC drawcircle surrounding thing Timothy Herchen henry gunn Honors Peter Herreshoff margincm Copyright co Arastradero Road Palo Alto CA USA pherreshoffpausdorg who draft several chapters particularly fellow teachers Dave Deggeller Danny Hahn resurrecting pamphlet grade took retyping soft copy theft Gautam Mittal editing academic student assistant Maya Sankar her typesetting incorporating rewriting Brandon Chung Andrew Chang additional formatting typeset diagrams tikz Asymptote source code githubcomanematodegatm contributions welcome compiled eigenvector goes solid ticksTicks xmin xmax axisXEquals family roughly suspected lattice transformed lie near lies I'm superimposed guessandcheck terrible develop procedure xy leftmultiply assumed existthat's factors eigenvalue xyxy yx xysfor xys eigenvalueeigenvector interestingly acted confused drawexpithetaexpitheta expitheta xaxisx yaxis'y' referring completely pairs calculations agree graphs match relationships displaced opposites strange what's iixy ixyxiy ix si weird fascinating discriminant other's graphed ie simplifies sums characteristic polynomialThe determining further assuming conditions impose expanded greater bacpq Manipulate inequality restriction bacxyxy equivalently InequalityThe states bacur rust u ru ur great potential potentially squared wordy smaller equality sections ago shouldn't explicitly currentprojectionperspective triple drawOv rotz rotatetheta vp rotzv drawOvp zaxis nuttriple ax ay drawvnutv drawvpnutvp stationary minors knew invalid anyway anything distances axisNote axes considering righthanded performed combinations axisthough origincentered sphere movement srmmaplambda mtrx mtrxlstriprstripsplit ssplitrprint threemat joinitem sublist maplambda joinmaplambda commute faces facing leave requires visualization sd sdA sdB sdC H sdD drawABCDA drawHEF drawHGF drawAE drawBF drawCG drawDH position cntr drawshiftr cntrshiftr rsqrtcntrshift rcntr dotcntr figureRotation shiftr dotrotate disc drawGdisc careful righttoleft confirming dealt ArcArrow NNE eigenvectorsThe average Joe spent deal invertible tends clumsy practice virtue people recognizably much random you'll pick sameor directly oppositedirection linearly vectors' helpful representation distributive generic Greek letter lambda traditionally gave place turns relatively intuition pointsThese antipodes glossary alignleft bijection bijective formally injective surjective binary acting cardinality rotational reflective symmetries polygon operated Euler's totient tells prime mathematically symbol generating generator entire series certain via group's output distinct codomain preserving preserves structure permutation arranged input shifted proportional pointed sometimes adjacency injection iso morphic isometric isomer isotonic isotropy isotope morph shape metamorphosis amorphous anthropomorphism morpheme orders bigger partner generates coprime merely lame fr paired structurepreserving nontrivial sizes constructed counterexample infinitely guesses shorthand made vs segment Cantor ask teacher please revise producing impossibility optional challenge straightforward subset state CantorSchrderBernstein surprisingly reduced ordered increasing classic Cantor's tough logically sound denoted interval reals chain correspondences straight forward versaAgain uses exists challenging portion interleave digits main outputs arbitrarily trailing zeroes numerically regime simplest observing noncollinear space expressed iteratively deleting open middle soThis literally down copying intuitively inter segmentslength segmentsj drawss interpushs interpush interlength interj interdelete immediately tell whether TERNARY consisting something choices continues infinitum inclusive asking heres nope fan oof none wonder odd analogously divisible uncountable countable axiom choiceThe indexed nonempty exactly item candidate toughIn explicit consistent 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forth posts snapped configuration regardless remove drawelems connects xx xz showed listing subgroupsHere guess meets criteria upon fourpost connections Systematically labels drawSnapElementsi figureElements getTLint rem quot mfw TLcorner getTLi drawSnapElementsTLcorner creating massive figureGroup shortcut program fivepost characteristics determine arise split completed gone glance ys ye achievable per harder common builder maximum Landau's treating extends alightergrayc itertools perms listIABCDEFGHJKLMNOPQRSTUVWX def snapelem elemelemn namefromelem testelem elementskey perm enumerateperms elementslabelsi print todraw standardindicesi topbrace brace drawtopbrace Color white startingelements doubled Phantom stuff whitebp strings ties combinatorics theory studied basis topics unfamiliar containing convince yourself switches pairings anchored looping looped touching stacked configurations reads somewhat terminology threepost countless precise talk subscript unsurprisingly objects study somename subgroupsWhat drawpointbetween pointbetween figureMatrix dotCp drawpointBBp pointBBp drawpointCCp pointCCp operating includes ratio less summing determined Pythagorean drawApBpCpcycle ApBp BpCp CpAp conclude quantities consolidating performing dotDp discontinuous belong applies collinearity spaced off ended appropriate gonna favorite representable vertically informally axbmxk axh bmxhk dmxk cxhdmxhk ah bmh ch dmh bmxk cxh dmxhk devil's coincidentthat they're equalthen formulaor justificationto subproblems I'll fixedits preimagewhen mundane subproblem plugging accurately separate LeftTicks RightTicks figurePreimage YEquals ymin fully classify mentioned quantify eg absolutely soon neat sort trick drawUnitSquareTransformreal dirA dirB dirC dirD drawApBpCpDpAp originaldirs imagedirs unitunitApDpunitApBp unitunitBpCpunitBpAp unitunitCpDpunitCpBp unitunitDpCpunitDpAp originalnames imageslabeledalready originalp originali originald originaldirsi originaln originalnamesi extranames lengthimagejoriginali enough extranamespushoriginalnamesj ' imageslabeledalreadyj labelstr labellabelstr imageslabeledalreadyi extra lengthimagejimagei originalnamesj imagei imagedirsi drawUnitSquareTransform versus parallelograms knowing constraints learn tackle properly sqf Et drawStEt drawUnitSquareTransformsqf drawUnitSquareTransformsqrt unchanged blanks translates leaving chosen translation translations translate hints comfortable allowed operators Cartesian option matrixa consistency operates mpproblemi matrixset picked generally keeps kinds glide representations investigating anothername figureProblem shearYou heard wind velocity scissors exert cut quantitatively sheared horizontally shoving changing nonzeroIf limitation obviously threedimensional twodimensional preimages App Bpp Cpp Dpp ABCDcycle ApBpCpDpcycle Spp AppBppCppDppcycle drawsurfaceS drawsurfaceSp drawsurfaceSpp drawSp drawSpp dotApp dotBpp dotCpp dotDpp Tu drawATuAp drawBTuBp drawCTuCp drawDTuDp drawApTuApp 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methodology face camera adjacent positions aforementioned continuing permutationThis haven't Nah too imo matrices' resemblance trickier reducible luckily naming ordersNot realm restrictive leap bear moment looked carefully volleyball brag funny memory PE someoneI whohow little feature seams directional xproj yproj zproj pointp onesixth succulentlyDrawpath fracalong seg fracalongpointp xprojptx yprojpty zprojptz drawseg dottedgray drawSurroundingBoundary onhorizon unitxproj currentprojectionorthographicxproj succulentlyDrawtransformsip striations rt succulentlyDrawtransformsirtp drawshiftcntrr figureThis restricted motivation analyzing accept flipping midplane vertexvertex midplaneboundary intersectionc drawsurfacemidplaneboundaryintersectionc evenoddopacitygray drawmidplaneboundary drawintersectionc opacitydotted cntrish arcvert drawarcvertrotate cntrisharcvertrotate cntrisharcvert dotarcvert dotv concrete oriented choicesnot fourfor bilateral fourfold mirror multiplies volleyballlet's 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lccccccccc Flour Starter Yeast Water Salt Soda Sugar Butter wants ingredients sight doubles leaves perspective holds rules abicdi acbd bcad adbc adbci negating tr BL parabola parametric graphing xreal tt yreal drawgraphx BottomTop LeftRight RightTickstrailingzero ingenuity calculus learned guessed connection alarmed seconds fairly lengthy revisit topic toward normal chemistry nerds worlds allows shapes hexagons informal painted rect Pouter Pinner lp drawrect filldrawPouter filldrawPinner drawtrrect filldrawtrPouter filldrawtrPinner AMBULATES rectangle's smallest groupThere stated soA soB soC soD figureRectangle singleton comprised consists labl frr frf frfrrffr abABB'A'DCC'D' bbanama baa babaabba ban aman amcPDCD'C'ABB'A bDdPdcbadZa bdaxbyddaxbyaddabddbG mc zzzn gngn zzdzzzdzzz DNDN dn bxay bxayx zzzga cGa dG irrrrrrrrr labelsi prevpt drawpentagonvertices drawshifti rcosang sinangrcosang sinang verticesinsert verticesdelete pentagons wear helmets lest damage pentagonal prism dependence akin horizontal asserting workIf created pyramid tetrahedron triangular vertexlet's midpoints enumerated drawaxisstring stretchastretchb stretchbstretcha drawstartend currentprojectionorthographic drawBCDA drawDB drawAC drawaxisvA BCD drawaxisvB ACD drawaxisvC ABD drawaxisvD drawaxise cows dotcowsicowsj dotcowscowscowscowscowsi figureRegular tetrahedron's viewed moving verticesyou dotBCD chaining aI bIa baab baba babba babab bab successfully holding twisting tricky dotDC dotAB drawBCD pointDA drawDinter drawinterA pointCA drawCinter drawmidplane drawsurfacemidplane minimal drawCubepicture drawpic ABCCpBpApABBp ADC ApDpD DpCp labelslength labelpic drawACp drawBDp drawApC drawBpD drawCubecurrentpicture figureMarking spacediagonally saying verified higherlevel placed uniquely determines closest furthest demonstrate typelist namelist drawsqpair loc ls shiftloc drawsqx typelisti labelnamelisti labelYZ labelXZ labelXY labelings demonstrated creates incredible cycling forwardand elementsgenerate drawcr crsqrt rsqrt cr naturally reflected version mine mathSE user verret concepts inaccessible permuting cycled permute permutes Lagrange's labeling A' B' C' D' solely socalled rotoreflection rrgproblemi drawTrianglepair vstretch bool showaxes false shownaxes labelaxes labelorder drawExtLinepair unitifyA axisnames shownaxesi verticesi axisname axisnamesi verticesj drawExtLinevertex drawTriangleoffset subgroupthe rotationsis twomember got remaining rotationeven unmirrored counterpart drawTriangle basealign member currentprojection drawDEFcycle drawBE drawCF unitifyDtriple pz drawDExtLinetriple succ drawCD unitifyDA drawDExtLineCF drawDExtLineAD CE drawDExtLineBE curved circular splinely interpolating ARQUE cosine stays sine arrowptcount arrowradius xpusharrowradius ypush zpusharrowradius currentlight prism's coplanar paperthin spatial dimensionssomething par excellence exchange achieved notebook 's' topleftbottomright homomorphism nominally shape's endpoints drawshift generatetetrahedron ret mid jutting peak centroid drawtetratriple drawverticesiverticesj orthographic tetra drawtetratetra newtetra newtetrai drawtetranewtetra tetratetra rotmc drawaxis dotaxis unitifytriple sqrtdxdxdydydzdz rotcenter rcsdv deviant rots rote roti drawrotsrotirote centerpath centercenter drawpointcenterpath pointcenterpath labelYZRotate simplex faithfully drawtetravertices drawEverticesi dotverticesi dotE realization expressible figureUnfilled alternate figureCompleted essence threeelement resolve twoelement oneelement arguments exponentiation bitwise convolution labelnames RTL evaluation unitsizecm radius cisangle drawcenter EndArrow labeli ordersi labelnamesi ending cardboard triangle's sequencing sided di hedral etymology refers polygonal pentagon's cube's enumLast groups' twopost onepost trigonometry Quinoa ram drawram iram rotateatan drawiram cbrace braceA drawcbrace drawrotateatan FshiftFram conservation nested atriangles Asmall Abig Atriangle aabb QED Similarity 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