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Matrices vt transformationes plani

Svmmarivm

Haec lectio omnem matricem dvorvm versvvm et dvarvm colvmnarvm interpretatvr vt transformationem plani, eam scilicet qvae qvodqve v⃗\vec{v}, vt colvmnam scriptvm, in vectorem Av⃗A\vec{v} transfert. Ostendit has transformationes esse ipsas lineares et qvamqve determinari imaginibvs dvorvm vectorvm basis canonicae, qvae svnt colvmnae matricis eivs. Hoc principio matrices invenit conversionvm, proiectionvm in rectas per originem transevntes et reflexionvm in eisdem rectis, homothetiarvm et obliqvationvm. Demonstrat transformationes componere idem esse ac matrices earvm mvltiplicare, vnde seqvvntvr additio angvlorvm dvarvm conversionvm, id qvod dvae reflexiones compositae conversionem efficivnt angvli dvpli eivs qvi inter rectas earvm est, et id qvod reflexiones generatim non commvtantvr. Determinans interpretatvr vt factorem qvo transformatio areas mvltiplicat, cvm signo negativo vbi orientationem invertit, et transformationes biiectivas describit vt eas qvarvm determinans nvllvm non est. Finit rotationem axivm aeqvationis generalis secvndi gradvs forma matriciali rescribens.

Proposita discendi

  1. Matricem vt transformationem linearem plani agnoscere et matricem transformationis ex imaginibvs vectorvm basis canonicae invenire.
  2. Matrices conversionvm, proiectionvm, reflexionvm, homothetiarvm et obliqvationvm scribere et dedvcere, et effectvm matricis datae geometrice describere.
  3. Demonstrare compositionem transformationvm prodvcto matricvm respondere, idqve ad compositionem conversionvm et reflexionvm adhibere.
  4. Determinans vt factorem areae et orientationis interpretari, diivdicare qvando transformatio invertibilis sit, et rotationem axivm aeqvationis secvndi gradvs matricibvs exprimere.

Matrices vt transformationes

In lectione cvrsvs Algebra et Geometria I de vectoribvs in plano vector definitvs est vt par ordinatvm nvmerorvm realivm, et in lectione de matricibvs et operationibvs cvm matricibvs definitvm est prodvctvm matricis per aliam, versvs in colvmnam. Si par (x,y)(x, y) vt colvmna scribitvr, id est vt matrix dvorvm versvvm et vnivs colvmnae, matrix AA dvorvm versvvm et dvarvm colvmnarvm id mvltiplicare potest, et qvod efficitvr alia colvmna dvorvm elementorvm est, id est alivs vector. Hoc modo qvaeqve matrix pvncta plani movet, et algebra matricvm significationem geometricam acqvirit. Posthac par (x,y)(x, y) cvm colvmna elementorvm xx et yy identificamvs, et e⃗1=(1,0)\vec{e}_1 = (1, 0) et e⃗2=(0,1)\vec{e}_2 = (0, 1) vectores basis canonicae appellamvs, ita vt omnis vector scribatvr v⃗=xe⃗1+ye⃗2\vec{v} = x\vec{e}_1 + y\vec{e}_2.

Transformatio plani est fvnctio qvae cviqve vectori plani alivm vectorem plani attribvit. Si AA est matrix dvorvm versvvm et dvarvm colvmnarvm, transformatio matricis AA ea est qvae cviqve v⃗\vec{v} prodvctvm Av⃗A\vec{v} attribvit.
T(v⃗)=Av⃗=(abcd)(xy)=(ax+bycx+dy)T(\vec{v}) = A\vec{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}

Exempli gratia, matrix elementorvm 22, 00, 00 et 33 pvnctvm (x,y)(x, y) in pvnctvm (2x,3y)(2x, 3y) transfert: planvm in directione horizontali ad dvplvm extendit et in verticali ad triplvm. Transformationes matricis proprietatem habent qvae eas a ceteris fvnctionibvs plani distingvit, qvaeqve definitione infra posita exprimitvr.

Transformatio TT plani linearis est si svmmam vectorvm et prodvctvm vectoris per nvmervm servat: pro omnibvs u⃗\vec{u} et v⃗\vec{v} plani et omni nvmero reali tt dvae aeqvalitates infra positae valent.
T(u⃗+v⃗)=T(u⃗)+T(v⃗),T(tv⃗)=tT(v⃗)T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v}), \qquad T(t\vec{v}) = tT(\vec{v})
TheoremaColvmnae matricis

Omnis transformatio matricis linearis est, et colvmnae matricis eivs svnt imagines vectorvm basis canonicae: prima est imago ipsivs e⃗1\vec{e}_1, altera ipsivs e⃗2\vec{e}_2. Vicissim, omnis transformatio linearis plani est transformatio vnivs matricis, eivs scilicet qvae colvmnas habet T(e⃗1)T(\vec{e}_1) et T(e⃗2)T(\vec{e}_2).

Adhibentvr distribvtivitas prodvcti matricvm et eivs congrventia cvm mvltiplicatione per nvmervm, in lectione de matricibvs et earvm operationibvs demonstratae. Figvra transformationem matricis AA colvmnarvm (2,1)(2, 1) et (−1,1)(-1, 1) seqvitvr.

Demonstratio

  1. T(u⃗+v⃗)=A⋅(u⃗+v⃗)=Au⃗+Av⃗,T(tv⃗)=A⋅(tv⃗)=tAv⃗T(\vec{u} + \vec{v}) = \resaltar{A \cdot \left( \vec{u} + \vec{v} \right)} = \resaltar{A\vec{u} + A\vec{v}}, \qquad T(t\vec{v}) = \resaltar{A \cdot \left( t\vec{v} \right)} = \resaltar{tA\vec{v}}

    Sit AA matrix et TT eivs transformatio. Distribvtivitas et congrventia cvm mvltiplicatione per nvmervm dvas condiciones linearitatis praebent; in figvra, parallelogrammvm qvod u⃗\vec{u} et v⃗\vec{v} efficivnt in id transfertvr qvod imagines eorvm efficivnt.

  2. T(e⃗1)=(abcd)(10)=(ac)T(\vec{e}_1) = \resaltar{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}} = \dato{c1}{\resaltar{\begin{pmatrix} a \\ c \end{pmatrix}}}

    Imaginem ipsivs e⃗1\vec{e}_1 compvtamvs: versvs in colvmnam mvltiplicando, ex qvoqve versv ipsivs AA solvm primvm elementvm valet.

  3. T(e⃗2)=(abcd)(01)=(bd)T(\vec{e}_2) = \resaltar{\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix}} = \dato{c2}{\resaltar{\begin{pmatrix} b \\ d \end{pmatrix}}}

    Eodem modo, in imagine ipsivs e⃗2\vec{e}_2 solvm altervm elementvm cvivsqve versvs valet.

  4. T(v⃗)=T(xe⃗1+ye⃗2)=xT(e⃗1)+yT(e⃗2)=xAe⃗1+yAe⃗2=Av⃗T(\vec{v}) = \resaltar{T(x\vec{e}_1 + y\vec{e}_2)} = xT(\vec{e}_1) + yT(\vec{e}_2) = \resaltar{xA\vec{e}_1 + yA\vec{e}_2} = A\vec{v}

    Sit nvnc TT transformatio linearis qvaelibet, et sit AA matrix cvivs colvmnae svnt T(e⃗1)T(\vec{e}_1) et T(e⃗2)T(\vec{e}_2). Scribimvs v⃗=xe⃗1+ye⃗2\vec{v} = x\vec{e}_1 + y\vec{e}_2 et linearitatem adhibemvs; per dvos gradvs svpra positos, T(e⃗1)=Ae⃗1T(\vec{e}_1) = A\vec{e}_1 et T(e⃗2)=Ae⃗2T(\vec{e}_2) = A\vec{e}_2. In figvra, reticvlvm plani in reticvlvm parallelogrammorvm transfertvr.

  5. Be⃗1=T(e⃗1)=Ae⃗1,Be⃗2=T(e⃗2)=Ae⃗2→B=AB\vec{e}_1 = T(\vec{e}_1) = A\vec{e}_1, \quad B\vec{e}_2 = T(\vec{e}_2) = A\vec{e}_2 \rightarrow B = A

    Si alia matrix BB qvoqve T(v⃗)=Bv⃗T(\vec{v}) = B\vec{v} pro omni v⃗\vec{v} impleret, colvmnae ipsivs BB, qvae svnt Be⃗1B\vec{e}_1 et Be⃗2B\vec{e}_2, cvm eis ipsivs AA congrverent; matrix vnica est.

Theorema stvdivm transformationis linearis ad stvdivm dvorvm vectorvm redigit: vt matrix cognoscatvr, satis est scire qvo eant e⃗1\vec{e}_1 et e⃗2\vec{e}_2, et vt sciatvr qvid matrix faciat, satis est colvmnas eivs legere. Ex linearitate praeterea complvres proprietates geometricae seqvvntvr. Origo fixa manet, qvia A0⃗=0⃗A\vec{0} = \vec{0}; ideo translationes, qvae originem movent, lineares non svnt. Recta aeqvationis vectorialis p⃗+td⃗\vec{p} + t\vec{d}, in lectione cvrsvs I de vectoribvs tractata, in copiam pvnctorvm Ap⃗+tAd⃗A\vec{p} + tA\vec{d} transfertvr, qvae recta est si Ad⃗A\vec{d} nvllvs non est, et vnvm pvnctvm si nvllvs est. Eodem modo segmentvm in segmentvm vel in pvnctvm transfertvr, rectae parallelae in rectas parallelas vel in pvncta, et qvadratvm verticvm 0⃗\vec{0}, e⃗1\vec{e}_1, e⃗1+e⃗2\vec{e}_1 + \vec{e}_2 et e⃗2\vec{e}_2, qvod qvadratvm vnitarivm appellabimvs, in parallelogrammvm transfertvr, fortasse complanatvm, qvod dvae colvmnae matricis efficivnt.