Skip to main content Contents Prev Up Next \(\def\vs{{\it vs. }}
\def\cf{{\it cf. }}
\def\viz{{\it viz. }}
\def\ie{{\it i.e. }}
\def\etc{{\it etc. }}
\def\eg{{\it e.g. }}
\def\etal{{\it et al .}}
\def\via{{\it via }}
\def\adhoc{{\it ad hoc }}
\def\apriori{{\it apriori }}
\def\Afrak{\mathfrak{A}}
\def\Bfrak{\mathfrak{B}}
\def\Cfrak{\mathfrak{C}}
\def\Dfrak{\mathfrak{D}}
\def\Efrak{\mathfrak{E}}
\def\Ffrak{\mathfrak{F}}
\def\Gfrak{\mathfrak{G}}
\def\Hfrak{\mathfrak{H}}
\def\Ifrak{\mathfrak{I}}
\def\Jfrak{\mathfrak{J}}
\def\Kfrak{\mathfrak{K}}
\def\Lfrak{\mathfrak{L}}
\def\Mfrak{\mathfrak{M}}
\def\Nfrak{\mathfrak{N}}
\def\Ofrak{\mathfrak{O}}
\def\Pfrak{\mathfrak{P}}
\def\Qfrak{\mathfrak{Q}}
\def\Rfrak{\mathfrak{R}}
\def\Sfrak{\mathfrak{S}}
\def\Tfrak{\mathfrak{T}}
\def\Ufrak{\mathfrak{U}}
\def\Vfrak{\mathfrak{V}}
\def\Wfrak{\mathfrak{W}}
\def\Xfrak{\mathfrak{X}}
\def\Yfrak{\mathfrak{Y}}
\def\Zfrak{\mathfrak{Z}}
\def\afrak{\mathfrak{a}}
\def\bfrak{\mathfrak{b}}
\def\cfrak{\mathfrak{c}}
\def\dfrak{\mathfrak{d}}
\def\efrak{\mathfrak{e}}
\def\ffrak{\mathfrak{f}}
\def\gfrak{\mathfrak{g}}
\def\hfrak{\mathfrak{h}}
\def\ifrak{\mathfrak{i}}
\def\jfrak{\mathfrak{j}}
\def\kfrak{\mathfrak{k}}
\def\lfrak{\mathfrak{l}}
\def\mfrak{\mathfrak{m}}
\def\nfrak{\mathfrak{n}}
\def\ofrak{\mathfrak{o}}
\def\pfrak{\mathfrak{p}}
\def\qfrak{\mathfrak{q}}
\def\rfrak{\mathfrak{r}}
\def\sfrak{\mathfrak{s}}
\def\tfrak{\mathfrak{t}}
\def\ufrak{\mathfrak{u}}
\def\vfrak{\mathfrak{v}}
\def\wfrak{\mathfrak{w}}
\def\xfrak{\mathfrak{x}}
\def\yfrak{\mathfrak{y}}
\def\zfrak{\mathfrak{z}}
\def\AA{\mathbb{A}}
\def\BB{\mathbb{B}}
\def\CC{\mathbb{C}}
\def\DD{\mathbb{D}}
\def\EE{\mathbb{E}}
\def\FF{\mathbb{F}}
\def\GG{\mathbb{G}}
\def\HH{\mathbb{H}}
\def\II{\mathbb{I}}
\def\JJ{\mathbb{J}}
\def\KK{\mathbb{K}}
\def\LL{\mathbb{L}}
\def\MM{\mathbb{M}}
\def\NN{\mathbb{N}}
\def\OO{\mathbb{O}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\SS{\mathbb{S}}
\def\TT{\mathbb{T}}
\def\UU{\mathbb{U}}
\def\VV{\mathbb{V}}
\def\WW{\mathbb{W}}
\def\XX{\mathbb{X}}
\def\YY{\mathbb{Y}}
\def\ZZ{\mathbb{Z}}
\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calC{\mathcal{C}}
\def\calD{\mathcal{D}}
\def\calE{\mathcal{E}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calH{\mathcal{H}}
\def\calI{\mathcal{I}}
\def\calJ{\mathcal{J}}
\def\calK{\mathcal{K}}
\def\calL{\mathcal{L}}
\def\calM{\mathcal{M}}
\def\calN{\mathcal{N}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calQ{\mathcal{Q}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\calU{\mathcal{U}}
\def\calV{\mathcal{V}}
\def\calW{\mathcal{W}}
\def\calX{\mathcal{X}}
\def\calY{\mathcal{Y}}
\def\calZ{\mathcal{Z}}
\def\Ap{A^\prime}
\def\Bp{B^\prime}
\def\Cp{C^\prime}
\def\Dp{D^\prime}
\def\Ep{E^\prime}
\def\Fp{F^\prime}
\def\Gp{G^\prime}
\def\Hp{H^\prime}
\def\Ip{I^\prime}
\def\Jp{J^\prime}
\def\Kp{K^\prime}
\def\Lp{L^\prime}
\def\Mp{M^\prime}
\def\Mp{N^\prime}
\def\Op{O^\prime}
\def\Pp{P^\prime}
\def\Qp{Q^\prime}
\def\Rp{R^\prime}
\def\Sp{S^\prime}
\def\Tp{T^\prime}
\def\Up{U^\prime}
\def\Vp{V^\prime}
\def\Wp{W^\prime}
\def\Xp{X^\prime}
\def\Yp{Y^\prime}
\def\Zp{Z^\prime}
\def\ap{a^\prime}
\def\bp{b^\prime}
\def\cp{c^\prime}
\def\dprime{d^\prime}
\def\ep{e^\prime}
\def\fp{f^\prime}
\def\gp{g^\prime}
\def\hp{h^\prime}
\def\ip{i^\prime}
\def\jp{j^\prime}
\def\kp{k^\prime}
\def\lp{l^\prime}
\def\mp{m^\prime}
\def\np{n^\prime}
\def\op{o^\prime}
\def\pp{p^\prime}
\def\qp{q^\prime}
\def\rp{r^\prime}
\def\sp{s^\prime}
\def\tp{t^\prime}
\def\up{u^\prime}
\def\vp{v^\prime}
\def\wp{w^\prime}
\def\xp{x^\prime}
\def\yp{y^\prime}
\def\zp{z^\prime}
\def\App{A^{\prime\prime}}
\def\Bpp{B^{\prime\prime}}
\def\Cpp{C^{\prime\prime}}
\def\Dpp{D^{\prime\prime}}
\def\Epp{E^{\prime\prime}}
\def\Fpp{F^{\prime\prime}}
\def\Gpp{G^{\prime\prime}}
\def\Hpp{H^{\prime\prime}}
\def\Ipp{I^{\prime\prime}}
\def\Jpp{J^{\prime\prime}}
\def\Kpp{K^{\prime\prime}}
\def\Lpp{L^{\prime\prime}}
\def\Mpp{M^{\prime\prime}}
\def\Mpp{N^{\prime\prime}}
\def\Opp{O^{\prime\prime}}
\def\Ppp{P^{\prime\prime}}
\def\Qpp{Q^{\prime\prime}}
\def\Rpp{R^{\prime\prime}}
\def\Spp{S^{\prime\prime}}
\def\Tpp{T^{\prime\prime}}
\def\Upp{U^{\prime\prime}}
\def\Vpp{V^{\prime\prime}}
\def\Wpp{W^{\prime\prime}}
\def\Xpp{X^{\prime\prime}}
\def\Ypp{Y^{\prime\prime}}
\def\Zpp{Z^{\prime\prime}}
\def\app{a^{\prime\prime}}
\def\bpp{b^{\prime\prime}}
\def\cpp{c^{\prime\prime}}
\def\dpp{d^{\prime\prime}}
\def\epp{e^{\prime\prime}}
\def\fpp{f^{\prime\prime}}
\def\gpp{g^{\prime\prime}}
\def\hpp{h^{\prime\prime}}
\def\ipp{i^{\prime\prime}}
\def\jpp{j^{\prime\prime}}
\def\kpp{k^{\prime\prime}}
\def\lpp{l^{\prime\prime}}
\def\mpp{m^{\prime\prime}}
\def\npp{n^{\prime\prime}}
\def\opp{o^{\prime\prime}}
\def\ppp{p^{\prime\prime}}
\def\qpp{q^{\prime\prime}}
\def\rpp{r^{\prime\prime}}
\def\spp{s^{\prime\prime}}
\def\tpp{t^{\prime\prime}}
\def\upp{u^{\prime\prime}}
\def\vpp{v^{\prime\prime}}
\def\wpp{w^{\prime\prime}}
\def\xpp{x^{\prime\prime}}
\def\ypp{y^{\prime\prime}}
\def\zpp{z^{\prime\prime}}
\def\abar{\overline{a}}
\def\bbar{\overline{b}}
\def\cbar{\overline{c}}
\def\dbar{\overline{d}}
\def\ebar{\overline{e}}
\def\fbar{\overline{f}}
\def\gbar{\overline{g}}
\def\ibar{\overline{i}}
\def\jbar{\overline{j}}
\def\kbar{\overline{k}}
\def\lbar{\overline{l}}
\def\mbar{\overline{m}}
\def\nbar{\overline{n}}
\def\obar{\overline{o}}
\def\pbar{\overline{p}}
\def\qbar{\overline{q}}
\def\rbar{\overline{r}}
\def\sbar{\overline{s}}
\def\tbar{\overline{t}}
\def\ubar{\overline{u}}
\def\vbar{\overline{v}}
\def\wbar{\overline{w}}
\def\xbar{\overline{x}}
\def\ybar{\overline{y}}
\def\zbar{\overline{z}}
\def\Abar{\overline{A}}
\def\Bbar{\overline{B}}
\def\Cbar{\overline{C}}
\def\Dbar{\overline{D}}
\def\Ebar{\overline{E}}
\def\Fbar{\overline{F}}
\def\Gbar{\overline{G}}
\def\Hbar{\overline{H}}
\def\Ibar{\overline{I}}
\def\Jbar{\overline{J}}
\def\Kbar{\overline{K}}
\def\Lbar{\overline{L}}
\def\Mbar{\overline{M}}
\def\Nbar{\overline{N}}
\def\Obar{\overline{O}}
\def\Pbar{\overline{P}}
\def\Qbar{\overline{Q}}
\def\Rbar{\overline{R}}
\def\Sbar{\overline{S}}
\def\Tbar{\overline{T}}
\def\Ubar{\overline{U}}
\def\Vbar{\overline{V}}
\def\Wbar{\overline{W}}
\def\Xbar{\overline{X}}
\def\Ybar{\overline{Y}}
\def\Zbar{\overline{Z}}
\def\aunder{\underline{a}}
\def\bunder{\underline{b}}
\def\cunder{\underline{c}}
\def\dunder{\underline{d}}
\def\eunder{\underline{e}}
\def\funder{\underline{f}}
\def\gunder{\underline{g}}
\def\hunder{\underline{h}}
\def\iunder{\underline{i}}
\def\junder{\underline{j}}
\def\kunder{\underline{k}}
\def\lunder{\underline{l}}
\def\munder{\underline{m}}
\def\nunder{\underline{n}}
\def\ounder{\underline{o}}
\def\punder{\underline{p}}
\def\qunder{\underline{q}}
\def\runder{\underline{r}}
\def\sunder{\underline{s}}
\def\tunder{\underline{t}}
\def\uunder{\underline{u}}
\def\vunder{\underline{v}}
\def\wunder{\underline{w}}
\def\xunder{\underline{x}}
\def\yunder{\underline{y}}
\def\zunder{\underline{z}}
\def\Aunder{\underline{A}}
\def\atilde{\widetilde{a}}
\def\btilde{\widetilde{b}}
\def\ctilde{\widetilde{c}}
\def\dtilde{\widetilde{d}}
\def\etilde{\widetilde{e}}
\def\ftilde{\widetilde{f}}
\def\gtilde{\widetilde{g}}
\def\htilde{\widetilde{h}}
\def\itilde{\widetilde{i}}
\def\jtilde{\widetilde{j}}
\def\ktilde{\widetilde{k}}
\def\ltilde{\widetilde{l}}
\def\mtilde{\widetilde{m}}
\def\ntilde{\widetilde{n}}
\def\otilde{\widetilde{o}}
\def\ptilde{\widetilde{p}}
\def\qtilde{\widetilde{q}}
\def\rtilde{\widetilde{r}}
\def\stilde{\widetilde{s}}
\def\ttilde{\widetilde{t}}
\def\utilde{\widetilde{u}}
\def\vtilde{\widetilde{v}}
\def\wtilde{\widetilde{w}}
\def\xtilde{\widetilde{x}}
\def\ytilde{\widetilde{y}}
\def\ztilde{\widetilde{z}}
\def\Atilde{\widetilde{A}}
\def\Btilde{\widetilde{B}}
\def\Ctilde{\widetilde{C}}
\def\Dtilde{\widetilde{D}}
\def\Etilde{\widetilde{E}}
\def\Ftilde{\widetilde{F}}
\def\Gtilde{\widetilde{G}}
\def\Htilde{\widetilde{H}}
\def\Itilde{\widetilde{I}}
\def\Jtilde{\widetilde{J}}
\def\Ktilde{\widetilde{K}}
\def\Ltilde{\widetilde{L}}
\def\Mtilde{\widetilde{M}}
\def\Ntilde{\widetilde{N}}
\def\Otilde{\widetilde{O}}
\def\Ptilde{\widetilde{P}}
\def\Qtilde{\widetilde{Q}}
\def\Rtilde{\widetilde{R}}
\def\Stilde{\widetilde{S}}
\def\Ttilde{\widetilde{T}}
\def\Utilde{\widetilde{U}}
\def\Vtilde{\widetilde{V}}
\def\Wtilde{\widetilde{W}}
\def\Xtilde{\widetilde{X}}
\def\Ytilde{\widetilde{Y}}
\def\Ztilde{\widetilde{Z}}
\def\Alphatilde{\widetilde{\Alpha}}
\def\Betatilde{\widetilde{\Beta}}
\def\Gammatilde{\widetilde{\Gamma}}
\def\Deltatilde{\widetilde{\Delta}}
\def\Epsilontilde{\widetilde{\Epsilon}}
\def\Zetatilde{\widetilde{\Zeta}}
\def\Etatilde{\widetilde{\Eta}}
\def\Thetatilde{\widetilde{\Theta}}
\def\Iotatilde{\widetilde{\Iota}}
\def\Kappatilde{\widetilde{\Kappa}}
\def\Lambdatilde{\widetilde{\Lamdba}}
\def\Mutilde{\widetilde{\Mu}}
\def\Nutilde{\widetilde{\Nu}}
\def\Xitilde{\widetilde{\Xi}}
\def\Omicrontilde{\widetilde{\Omicron}}
\def\Pitilde{\widetilde{\Pi}}
\def\Rhotilde{\widetilde{\Rho}}
\def\Sigmatilde{\widetilde{\Sigma}}
\def\Tautilde{\widetilde{\Tau}}
\def\Upsilontilde{\widetilde{\Upsilon}}
\def\Phitilde{\widetilde{\Phi}}
\def\Chitilde{\widetilde{\Chi}}
\def\Psitilde{\widetilde{\Psi}}
\def\Omegatilde{\widetilde{\Omega}}
\def\alphatilde{\widetilde{\alpha}}
\def\betatilde{\widetilde{\beta}}
\def\gammatilde{\widetilde{\gamma}}
\def\deltatilde{\widetilde{\delta}}
\def\epsilontilde{\widetilde{\epsilon}}
\def\zetatilde{\widetilde{\zeta}}
\def\etatilde{\widetilde{\eta}}
\def\thetatilde{\widetilde{\theta}}
\def\iotatilde{\widetilde{\iota}}
\def\kappatilde{\widetilde{\kappa}}
\def\lambdatilde{\widetilde{\lamdba}}
\def\mutilde{\widetilde{\mu}}
\def\nutilde{\widetilde{\nu}}
\def\xitilde{\widetilde{\xi}}
\def\omicrontilde{\widetilde{\omicron}}
\def\pitilde{\widetilde{\pi}}
\def\rhotilde{\widetilde{\rho}}
\def\sigmatilde{\widetilde{\sigma}}
\def\tautilde{\widetilde{\tau}}
\def\upsilontilde{\widetilde{\upsilon}}
\def\phitilde{\widetilde{\phi}}
\def\chitilde{\widetilde{\chi}}
\def\psitilde{\widetilde{\psi}}
\def\omegatilde{\widetilde{\omega}}
\def\Alphabar{\bar{\Alpha}}
\def\Betabar{\bar{\Beta}}
\def\Gammabar{\bar{\Gamma}}
\def\Deltabar{\bar{\Delta}}
\def\Epsilonbar{\bar{\Epsilon}}
\def\Zetabar{\bar{\Zeta}}
\def\Etabar{\bar{\Eta}}
\def\Thetabar{\bar{\Theta}}
\def\Iotabar{\bar{\Iota}}
\def\Kappabar{\bar{\Kappa}}
\def\Lambdabar{\bar{\Lamdba}}
\def\Mubar{\bar{\Mu}}
\def\Nubar{\bar{\Nu}}
\def\Xibar{\bar{\Xi}}
\def\Omicronbar{\bar{\Omicron}}
\def\Pibar{\bar{\Pi}}
\def\Rhobar{\bar{\Rho}}
\def\Sigmabar{\bar{\Sigma}}
\def\Taubar{\bar{\Tau}}
\def\Upsilonbar{\bar{\Upsilon}}
\def\Phibar{\bar{\Phi}}
\def\Chibar{\bar{\Chi}}
\def\Psibar{\bar{\Psi}}
\def\Omegabar{\bar{\Omega}}
\def\alphabar{\bar{\alpha}}
\def\betabar{\bar{\beta}}
\def\gammabar{\bar{\gamma}}
\def\deltabar{\bar{\delta}}
\def\epsilonbar{\bar{\epsilon}}
\def\zetabar{\bar{\zeta}}
\def\etabar{\bar{\eta}}
\def\thetabar{\bar{\theta}}
\def\iotabar{\bar{\iota}}
\def\kappabar{\bar{\kappa}}
\def\lambdabar{\bar{\lamdba}}
\def\mubar{\bar{\mu}}
\def\nubar{\bar{\nu}}
\def\xibar{\bar{\xi}}
\def\omicronbar{\bar{\omicron}}
\def\pibar{\bar{\pi}}
\def\rhobar{\bar{\rho}}
\def\sigmabar{\bar{\sigma}}
\def\taubar{\bar{\tau}}
\def\upsilonbar{\bar{\upsilon}}
\def\phibar{\bar{\phi}}
\def\chibar{\bar{\chi}}
\def\psibar{\bar{\psi}}
\def\omegabar{\bar{\omega}}
\def\del{\partial}
\def\delbar{\overline{\partial}}
\def\Cech{\check{C}}
\def\half{\frac{1}{2}}
\def\defeq{\mathrel{\mathop:}=}
\def\alg{\mathrm{alg}}
\def\Alt{\mathrm{Alt}}
\def\Amp{\mathrm{Amp}}
\def\Arg{\mathrm{Arg}}
\def\an{\mathrm{an}}
\def\anti{\mathrm{anti}}
\def\Ap{\mathrm{Ap}}
\def\arcsinh{\mathrm{arcsinh\hspace{0.07cm}}}
\def\arccosh{\mathrm{arccosh\hspace{0.07cm}}}
\def\arctanh{\mathrm{arctanh\hspace{0.07cm}}}
\def\arccsch{\mathrm{arccsch\hspace{0.07cm}}}
\def\arcsech{\mathrm{arcsech\hspace{0.07cm}}}
\def\arccoth{\mathrm{arccoth\hspace{0.07cm}}}
\def\arccsc{\mathrm{arccsc\hspace{0.07cm}}}
\def\arcsec{\mathrm{arcsec\hspace{0.07cm}}}
\def\arccot{\mathrm{arccot\hspace{0.07cm}}}
\def\arg{\mathrm{arg}}
\def\BC{\mathrm{BC}}
\def\Bel{\mathrm{Bel}}
\def\calCH{\mathcal{CH}}
\def\csch{\mathrm{csch}\hspace{0.07cm}}
\def\CH{\mathrm{CH}}
\def\ch{\mathrm{ch}}
\def\closed{\mathrm{closed}}
\def\codim{\mathrm{codim}}
\def\coth{\mathrm{coth}\hspace{0.07cm}}
\def\Coh{\mathfrak{Coh}}
\def\Coker{\mathrm{Coker}}
\def\Cone{\mathrm{Cone}}
\def\darg{d\mathrm{arg}}
\def\Db{\mathrm{Db}}
\def\dclosed{\mathrm{d-closed}}
\def\deg{\mathrm{deg}}
\def\dim{\mathrm{dim}}
\def\divisor{\mathrm{div}}
\def\dlog{d\mathrm{log}}
\def\DNE{\mathrm{DNE}}
\def\DR{\mathrm{DR}}
\def\DST{\mathrm{DST}}
\def\exp{\mathrm{exp}}
\def\FLB{\mathrm{FLB}}
\def\FLS{\mathrm{FLS}}
\def\Gr{\mathrm{Gr}}
\def\Hzar{H_{\mathrm{Zar}}}
\def\Hol{\mathrm{Hol}}
\def\Id{\mathrm{Id}}
\def\Image{\mathrm{Im}}
\def\Ka{\mathcal{K}_A}
\def\Ker{\mathrm{Ker}}
\def\kod{\mathrm{kod}}
\def\Kx{\mathcal{K}_X}
\def\Kz{\mathcal{K}_Z}
\def\log{\mathrm{log}}
\def\Log{\mathrm{Log}}
\def\Li{\mathrm{Li}}
\def\min{\mathrm{min}}
\def\Mon{\mathrm{Mon}}
\def\Nef{\mathrm{Nef}}
\def\NS{\mathrm{NS}}
\def\Oa{\mathcal{O}_A}
\def\Ox{\mathcal{O}_X}
\def\Oz{\mathcal{O}_Z}
\def\Perp{\mathrm{Perp}}
\def\Pic{\mathrm{Pic}}
\def\Proj{\mathrm{Proj}}
\def\rank{\mathrm{rank}}
\def\Rat{\mathrm{Rat}}
\def\Real{\mathrm{Re}}
\def\reg{\mathrm{reg}}
\def\Res{\mathrm{Res}}
\def\res{\mathrm{res}}
\def\Ric{\mathrm{Ric}}
\def\sech{\mathrm{sech}\hspace{0.07cm}}
\def\Span{\mathrm{Span}}
\def\Spec{\mathrm{Spec}}
\def\sing{\mathrm{sing}}
\def\Singx{\mathrm{Sing}(X)}
\def\sheafKer{\mathcal{\Ker}}
\def\sheafIm{\mathcal{\Im}}
\def\Span{\mathrm{Span}}
\def\Spin{\mathrm{Spin}}
\def\Str{\mathrm{Str}}
\def\td{\mathrm{td}}
\def\tr{\mathrm{tr}}
\def\Todd{\mathrm{Todd}}
\def\tor{\mathrm{tor}}
\def\trdeg{\mathrm{trdeg}}
\def\Zar{\mathrm{Zar}}
\def\ZFLS{\mathrm{ZFLS}}
\usepackage{tikz}
\usepackage{tkz-graph}
\usepackage{tkz-euclide}
\usetikzlibrary{patterns}
\usetikzlibrary{positioning}
\usetikzlibrary{matrix,arrows}
\usetikzlibrary{calc}
\usetikzlibrary{shapes}
\usetikzlibrary{through,intersections,decorations,shadows,fadings}
\usepackage{pgfplots}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 6.3 Examples in \(\RR^2\)
Subsection 6.3.1 Five Types
It is useful to look at transformations \(\RR^2 \rightarrow
\RR^2\) in order to build experience and intuition. As I described earlier, linear transformations preserve flat objects. What can be done to \(\RR^2\) that preserves lines and preserves the origin?
Proposition 6.3.1 .
There are five basic types of linear transformation of \(\RR^2\text{.}\)
Rotations about the origin (either clockwise or counter-clockwise) preserve lines. I can’t rotate around any other point since that would move the origin. Since I can choose any angle, there are infinitely many such rotations. Also, since rotating by \(\theta\) radians clockwise is the same as \(2\pi - \theta\) radians counter-clockwise, I typically choose to only deal with counter-clockwise rotations. Counter-clockwise rotations are considered positive and clockwise rotations are considered negative.
Reflections over lines through the origin preserve lines. The line of reflection must pass through the origin or else the reflection will move the origin. Since there are infinitely many different lines through the origin, there are infinitely many such reflections.
Skews are a little trickier to visualize. A skew is a transformation that takes either vertical or horizontal lines (but not both) and tilts them diagonally. It changes squares into parallelograms. The tilted lines are still lines, so it is a linear transformation.
Dilations are transformations which stretch or shrink in various directions.
Projections are transformations that collapse \(\RR^2\) down to a line through the origin. Two important examples are the projections onto either axis. Projection onto the \(x\) axis sends a point \((a,b)\) to \((a,0)\text{,}\) removing the \(y\) component entirely. Likewise, projection onto the \(y\) axis sends a point \((a,b)\) to \((0,b)\text{,}\) removing the \(x\) component. In a similar manner, I can project onto any line through the origin by sending each point to the closest point on the line. Finally, there is the projection to the origin which sends all points to the origin.
I present the following interesting theorem without proof.
Theorem 6.3.2 .
All linear transformations of \(\RR^2\) are generated by the composition of transformations of the previous five types.
Subsection 6.3.2 Matrices of the Five Types
The rotation in \(\RR^2\) by the angle \(\theta\) (counter-clockwise) around the origin given by the matrix
\begin{equation*}
\begin{pmatrix}
\cos \theta \amp - \sin \theta \\
\sin \theta \amp \cos \theta
\end{pmatrix}\text{.}
\end{equation*}
Following this formula, a quarter turn is
\begin{equation*}
\begin{pmatrix}
0 \amp -1 \\
1 \amp 0
\end{pmatrix}\text{.}
\end{equation*}
Likewise, a half turn is
\begin{equation*}
\begin{pmatrix}
-1 \amp 0 \\
0 \amp -1
\end{pmatrix}\text{.}
\end{equation*}
Finally, a three-quarter turn is
\begin{equation*}
\begin{pmatrix}
0 \amp 1 \\
-1 \amp 0
\end{pmatrix}\text{.}
\end{equation*}
If \(\begin{pmatrix} a \\ b \end{pmatrix}\) is a unit vector in \(\RR^2\text{,}\) then the reflection over the line in the direction of \(\begin{pmatrix} a \\ b
\end{pmatrix}\) is given by the matrix
\begin{equation*}
\begin{pmatrix}
a^2 - b^2 \amp 2ab \\
2ab \amp b^2 - a^2
\end{pmatrix}
\end{equation*}
Reflection over the \(x\) axis is
\begin{equation*}
\begin{pmatrix}
1 \amp 0 \\
0 \amp -1
\end{pmatrix}\text{.}
\end{equation*}
Reflection over the \(y\) axis is
\begin{equation*}
\begin{pmatrix}
- 1 \amp 0 \\
0 \amp 1
\end{pmatrix}\text{.}
\end{equation*}
Reflection over the the line \(y=x\) is
\begin{equation*}
\begin{pmatrix}
0 \amp 1 \\
1 \amp 0
\end{pmatrix}\text{.}
\end{equation*}
Reflection over the line \(y=-x\) is
\begin{equation*}
\begin{pmatrix}
0 \amp -1 \\
-1 \amp 0
\end{pmatrix}\text{.}
\end{equation*}
There isn’t necessarily an easily describable form for all skews. However, I can still give some examples. A horizontal skew that moves the positive \(y\) axis in the positive \(x\) direction is
\begin{equation*}
\begin{pmatrix}
1 \amp 0 \\
1 \amp 1
\end{pmatrix}\text{.}
\end{equation*}
Similarly, a skew in the positive \(y\) direction is
\begin{equation*}
\begin{pmatrix}
1 \amp 1 \\
0 \amp 1
\end{pmatrix}\text{.}
\end{equation*}
Dialations have two stretch factors: \(a\) in the \(x\) direction and \(b\) in the \(y\) direction. The matrix for this is
\begin{equation*}
\begin{pmatrix}
a \amp 0 \\
0 \amp b
\end{pmatrix}\text{.}
\end{equation*}
Again, let \(\begin{pmatrix} a \\ b \end{pmatrix}\) be a unit vector. This unit vector defines a line through the origin by taking all multiples of this vector. The projection onto this line is given by the matrix
\begin{equation*}
\begin{pmatrix}
a^2 \amp ab \\
ab \amp b^2
\end{pmatrix}\text{.}
\end{equation*}
In addition, there is one other projection: the projection to the origin. This sends everything to zero and its matrix is the zero matrix.
