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\pages{1-6}
\atxy{48}{254}{$PGL(2,\mathbb F)$ \href{https://planetmath.org/simplytransitive}{simply transitively} on the set of distinct ordered triples in $\mathbb{FP}^1$}
\pages{7-8}
\atxy{1}{116}{\small\parbox[t]{47mm}{Desargues illustrates a \href{https://en.wikipedia.org/wiki/3D_projection\#Perspective_projection}{perspective projection} defined except $P$.\\Choose point $P=[p]$ and hyperplane $\mathbb P(U)$,$$\mathbb P(\mathbb R^n)=P\oplus\mathbb P(U)$$So any $x$ can be written as$$x=p+u,u\in U$$we define the map\[\mathbb R^n\setminus\{P\}\to\mathbb P(U);\quad x\mapsto u\]See \href{https://graemewilkin.github.io/Geometry/Tiled_floor_proof.html}{2-point perspective drawing}}}
\pages{9-10}
\atxy{1}{155}{\parbox[t]{47mm}{Another `moduli problem'\\$GL(2,\mathbb C)$ acting on $M_{2,2}(\mathbb C)$ by conjugation $A:X\mapsto AXA^{-1}$ Numerical invariants:$$\det=\lambda_1\lambda_2,\text{trace}=\lambda_1+\lambda_2$$But different Jordan types even though det, trace same:\\$\begin{pmatrix}0&0\\0&0\end{pmatrix}\nsim\begin{pmatrix}0&\epsilon\\0&0\end{pmatrix}$ for $\epsilon\ne0$\\orbit not closed}}
\pages{11}
\atxy{1}{75}{$\begin{pmatrix}a_0&a_1&a_2\end{pmatrix}\begin{pmatrix}x_0\\x_1\\x_2\end{pmatrix}=0$}
\atxy{7.6}{80}{$\in V^*$}
\atxy{24}{84}{$\in V$}
\atxy{1}{90}{$\begin{pmatrix}a_0&a_1&a_2\end{pmatrix}$ defines a linear map $\mathbb F^3\to\mathbb F$}
\atxy{100}{144}{If $\dim V=\infty$, may be $(U_1\cap U_2)^\circ>U_1^\circ+U_2^\circ$}
\atxy{123}{75}{\parbox[t]{\textwidth}{If $e_1,\dots,e_n$ is a basis for $V$,\\dual basis $f_1,\dots,f_n$ for $V^*$ with $f_i(e_i)=\delta_{ij}$.}}
\pages{12}
\atxy{1}{98}{\parbox[t]{\textwidth}{Desargue:\\central perspective$\Rightarrow$axial perspective\\Desargue dual:\\central perspective$\Leftarrow$axial perspective}}
\atxy{170}{235}{$\begin{aligned}B(iv,iv)&=i^2B(v,v)\\&=-B(v,v)\end{aligned}$}
\atxy{97}{252}{$\Leftrightarrow B$ is invertible}
\atxy{1}{142}{\parbox[t]{47mm}{$B$ is nonsingular iff$$v\mapsto (x\mapsto B(x,v))$$is injective\\(In particular when $B$ is the inner product, it is the dual map $v\mapsto v^*$)\\For $\dim V$ finite, counting dimension shows it is an isomorphism, but for $\dim V=\infty$, it may not be surjective e.g.\ Dirac delta functional, see \href{https://en.wikipedia.org/wiki/Degenerate_bilinear_form}{Degenerate bilinear form}).\\}}
\pages{13}
\atxy{127}{49}{\begin{minipage}[t]{0.6\textwidth}\begin{flushright}$B(x,y)=\frac{Q(x+y)-Q(x-y)+i(Q(x+iy)-Q(x-iy))}4$\\defines a Hermitian form.\end{flushright}\end{minipage}}
\atxy{10}{80}{Sylvester's law of inertia}
\atxy{150}{139}{\parbox[t]{\textwidth}{$Q$ is determined by rank \#nonzero terms\\and\\signature $\#\text{square}-\#\text{negative squares}$}}
\pages{14}
\atxy{136}{90}{$\text{rank}=\#$nonzero eigenvalues}
\atxy{136}{95}{$s=\#$negative eigenvalues}
\atxy{1}{236}{\parbox[t]{47mm}{Every nonsingular nonempty conic on $\mathbb R$ can be put into the form $x^2+y^2=z^2$}}
\pages{15}
\atxy{1}{123}{\parbox[t]{44mm}{A {\bf manifold} is a topological space that is locally Euclidean.\\Thm. $f:\mathbb R^n\to\mathbb R$ the level set $f^{-1}(c)$ is a manifold of dim $n-1$ if $\forall x\in f^{-1}(c)$, the derivative $df_x:\mathbb R^n\xrightarrow{\text{linear}}\mathbb R$ has maximal rank.\\A cone $x^2+y^2-z^2=0$ is not a manifold:$$df=\begin{pmatrix}2x\\2y\\-2z\end{pmatrix}$$at vertex $df=0$.}}
\atxy{170}{158}{\parbox[t]{42mm}{Over $\mathbb R$:\\$x_0^2+x_1^2+x_2^2$ defines $\emptyset$\\Nonsingular nonempty conic is $x_0^2-x_1^2-x_2^2=0$\\This implies $x_0\ne0$\\so affinise$$1=\left(\frac{x_1}{x_0}\right)^2+\left(\frac{x_2}{x_0}\right)^2$$defines a circle in $\mathbb{RP}^2$\\if we remove instead $x_1=0$\\complement is$$\left(\frac{x_0}{x_1}\right)^2-\left(\frac{x_2}{x_1}\right)^2=1$$defines a hyperbola in $\mathbb{RP}^2$\\if we remove a tangent to conic\\Now complement is a parabola\\we recover the classical Greek `conic sections'}}
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\pages{16}
\atxy{1}{101}{\parbox[t]{47mm}{Over $\mathbb C$, this map $\alpha$ is a homeomorphism, so nonsingular conics in $\mathbb{CP}^2$ are topologically $S^2$.}}
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\atxy{179.5}{70}{\color{white}$x$}
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\atxy{172}{101}{\color{white}$y$}
\atxy{182}{86.7}{\color{white}$\alpha(y)$}
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\pages{17-18}
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