The Math

FoldUnfold Table of Contents Affine n-Space over a Field Affine n-Space over a Field Definition: Let $K$ be a field. The Affine $n$-Space over $K$ is the set $\mathbb{A}^n(K) = K^n$. The Points in $\mathbb{A}^n(K)$ are the elements of $\mathbb{A}^n(K)$. The notation “$K^n$” denotes the cartesian product $K^n = \underbrace{K \times K \times … \times […]

FoldUnfold Table of Contents The Generalized Fundamental Theorem of Projective Planes The Generalized Fundamental Theorem of Projective Planes Recall from The Fundamental Theorem of Projective Planes page that if $F$ is a field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $\mathbf{p}, \mathbf{q}, \mathbf{r}, \mathbf{s} \in \mathbb{P}^2(F)$ are points such that no three of them […]

FoldUnfold Table of Contents The Fundamental Theorem of Projective Planes The Fundamental Theorem of Projective Planes Theorem 1 (The Fundamental Theorem of Projective Planes): Let $\mathbf{z_1} = [1, 0, 0]$, $\mathbf{z_2} = [0, 1, 0]$, $\mathbf{z_3} = [0, 0, 1]$, and $\mathbf{z_4} = [1, 1, 1]$. If $\mathbf{p}, \mathbf{q}, \mathbf{r}, \mathbf{s}$ are four distinct points […]

FoldUnfold Table of Contents Product Collineations of Projective Planes Product Collineations of Projective Planes Let $F$ be a field and $\mathbb{P}^2(F)$ be the projective plane over $F$. Consider two $3 \times 3$ invertible matrices $M$ and $N$. As we have seen, we can define collineations on $\mathbb{P}^2(F)$ as functions $\phi_M, \phi_N : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ […]

FoldUnfold Table of Contents Inverse Collineations of Projective Planes Inverse Collineations of Projective Planes Recall from the Collineations of Projective Planes page that if $F$ is a field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $M$ is a $3 \times 3$ invertible matrix, then a collineation of $\mathbb{P}^2(F)$ is the bijective function $\phi_M : […]

FoldUnfold Table of Contents Identity Collineations of Projective Planes Identity Collineations of Projective Planes Recall from the Collineations of Projective Planes page that if $F$ is a field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $M$ is a $3 \times 3$ invertible matrix, then a collineation of $\mathbb{P}^2(F)$ is the bijective function $\phi_M : […]

FoldUnfold Table of Contents The Preservation of Collinear Points and Concurrent Lines with Collineations The Preservation of Collinear Points and Concurrent Lines with Collineations Recall from the Collineations of Projective Planes on Points page that if $F$ is any field, $\mathbb{P}^2(F)$ be the projective plane over $F$, and $M$ be any $3 \times 3$ invertible […]

FoldUnfold Table of Contents Collineations of Projective Planes on Lines Example 1 Collineations of Projective Planes on Lines Recall from the Collineations of Projective Planes on Points page that if $F$ is any field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $M$ is any $3 \times 3$ invertible matrix whose entries are from $F$ […]

FoldUnfold Table of Contents Collineations of Projective Planes on Points Collineations of Projective Planes on Points Like with transformations in the plane – we can define transformations on projective planes over a field $F$ which we define below. Definition: Let $F$ be any field and let $\mathbb{P}^2(F)$ be the projective plane over $F$. Let $M$ […]

FoldUnfold Table of Contents Projective Planes in the Finite Field Zp Projective Planes in the Finite Field Zp On The Fano Plane page we looked at the projective plane over the field of integers modulo $2$, $\mathbb{Z}_2$ known as the Fano Plane $\mathbb{P}^2 (\mathbb{Z}_2)$ – the smallest projective plane (due to $\mathbb{Z}_2$ being the smallest […]