Question

Let \(P=\left(a_{1}, \ldots, a_{n}\right), Q=\left(b_{1}, \ldots, b_{n}\right)\) be distinct points of \(\mathbb{A}^{n}\). The line through \(P\) and \(Q\) is defined to be \(\left.\left\\{a_{1}+t\left(b_{1}-a_{1}\right), \ldots, a_{n}+t\left(b_{n}-a_{n}\right)\right) \mid t \in k\right\\}\). (a) Show that if \(L\) is the line through \(P\) and \(Q\), and \(T\) is an affine change of coordinates, then \(T(L)\) is the line through \(T(P)\) and \(T(Q)\). (b) Show that a line is a linear subvariety of dimension 1, and that a linear subvariety of dimension 1 is the line through any two of its points. (c) Show that, in \(\mathbb{A}^{2}\), a line is the same thing as a hyperplane. (d) Let \(P\), \(P^{\prime} \in \mathbb{A}^{2}, L_{1}, L_{2}\) two distinct lines through \(P, L_{1}^{\prime}, L_{2}^{\prime}\) distinct lines through \(P^{\prime}\). Show that there is an affine change of coordinates \(T\) of \(\mathbb{A}^{2}\) such that \(T(P)=P^{\prime}\) and \(T\left(L_{i}\right)=L_{i}^{\prime}, i=1,2\).

Short answer

In summary, we have used the concepts of affine space, lines, affine change of coordinates, linear subvarieties, and hyperplanes to show the following: (a) When an affine change of coordinates is applied to a line, the transformed line remains the line between the transformed points P and Q. (b) A line is a linear subvariety of dimension 1, and a linear subvariety of dimension 1 can be represented by a line between any two of its points. (c) In 2-dimensional affine space, a line is the same as a hyperplane. (d) It is possible to find an affine change of coordinates that transforms points and lines according to the given conditions. By understanding and applying these concepts in various contexts, we can gain a better grasp of the properties and relationships between different geometric entities in affine space.

Step-by-step solution

Transformation of points P and Q

\(T(P) = AP + c\) and \(T(Q) = AQ + c\) Now, we transform the line \(L\) and call it \(L'\):

Transformation of the line L

\(L' = \{A(a_1 + t(b_1 - a_1)), \ldots, A(a_n + t(b_n - a_n))\} + c \mid t \in k\) To show that \(L'\) is the line through \(T(P)\) and \(T(Q)\), it suffices to show that:

Line L' passes through T(P) and T(Q)

\(L'(0) = AP + c\) and \(L'(1) = AQ + c\) By substituting \(t = 0\) and \(t = 1\) in the expression for \(L'\), we can see that these equalities hold, which concludes part (a). (b) Show that a line is a linear subvariety of dimension 1, and that a linear subvariety of dimension 1 is the line through any two of its points.

Line is a linear subvariety of dimension 1

A line has dimension 1 as it can be parametrized by a single parameter t. Furthermore, it is linear since all its components have a linear relationship with the parameter t.

Linear subvariety of dimension 1 is a line through two points

Given two points \(P\) and \(Q\) in that linear subvariety, we can define a line using the same parametrization as before, and this line will be included in the linear subvariety, as the relationship between any pair of its points and the parameter t is linear. (c) Show that, in \(\mathbb{A}^2\), a line is the same thing as a hyperplane.

Line is a hyperplane in A2

In \(\mathbb{A}^2\), a line and a hyperplane are both defined by a single linear equation. Thus, they are the same thing in this two-dimensional space. (d) Let \(P, P' \in \mathbb{A}^2, L_1, L_2\) two distinct lines through \(P, L_1', L_2'\) distinct lines through \(P'\). Show that there is an affine change of coordinates \(T\) of \(\mathbb{A}^2\) such that \(T(P) = P'\) and \(T(L_i) = L_i', i = 1,2\).

Translation of point P to P'

To map P to P', we can start by translating the coordinate system by the vector \((P' - P)\). Define \(T'(x) = x + (P' - P)\) and note that \(T'(P) = P'\).

Transformation for lines

Now, let's consider two new lines, \(L_1'' = T'(L_1)\) and \(L_2'' = T'(L_2)\). Observe that \(P' \in L_1''\) and \(P' \in L_2''\). We can now find a linear transformation \(T''\) that maps \(L_1''\) to \(L_1'\) and \(L_2''\) to \(L_2'\). To do that, we will define \(T''(x) = Ax\), where A is a 2x2 matrix. By solving the system of equations \(A(L_1'' - P') = L_1' - P'\) and \(A(L_2'' - P') = L_2' - P'\) for the entries of A, we can find a suitable matrix A. Finally, the desired affine change of coordinates is defined as:

Final affine change of coordinates

\(T(x) = Ax + (P' - P)\), which satisfies the required conditions.

Key concepts

Affine Transformations

Affine transformations are fundamental tools in affine geometry and are used to map one coordinate system to another. These transformations are composed of linear transformations followed by a translation.
An affine transformation of a point in \(\mathbb{A}^{n}\) can be expressed as \(T(P) = A \cdot P + c\), where \(A\) is an \(n \times n\) linear transformation matrix and \(c\) is a translation vector.

• The matrix \(A\) describes the linear part of the transformation, affecting the shape and orientation.
• The vector \(c\) introduces translation, shifting the position of points without rotating or scaling.

Affine transformations preserve lines and parallelism, which means if we transform a line, it remains a line in the new coordinate system. They are particularly useful for showing that geometrical properties are invariant under different coordinate systems.

Linear Subvarieties

A linear subvariety in affine geometry refers to a subset of a space that locally looks like a line, plane, or hyperplane, depending on the dimension. Specifically, a linear subvariety of dimension 1 is a line.
Every line in affine space can be described using a parameter \(t\), with coordinates expressed as a linear function of \(t\). For example, a line through points \(P\) and \(Q\) can be parameterized:\[ (1-t)P + tQ \text{ for } t \in \mathbb{R} \]

• This parameterization means the line is a collection of points that can be described continuously through \(t\).
• Any two distinct points in the linear subvariety can define such a line.

Linear subvarieties have the property that any combination of points with specific coefficients will still result in a point that lies within the subvariety. They reveal the algebraic structure of the geometry.

Algebraic Varieties

Algebraic varieties are central objects in algebraic geometry, which study solutions to systems of polynomial equations. Affine varieties, in particular, are defined as the set of common solutions to these polynomial equations in an affine space.
An algebraic variety can be of any dimension, and lines are one-dimensional examples of these varieties: they are solutions to linear equations.

• An affine variety in two dimensions can be a line, parabola, or any curve described by a polynomial equation.
• Varieties are significant because they generalize the notion of geometric shapes, providing a bridge between algebra and geometry.

While the concept of varieties becomes complex in higher dimensions and degrees, in the context of this exercise, focusing on linear subvarieties helps to understand basic properties like dimensionality and the interaction between points and lines.

Hyperplane

In affine geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. In \(\mathbb{A}^{2}\), for instance, a hyperplane is a line.
Hyperplanes can be defined linearly by a single equation, such as:\[ a_1x_1 + a_2x_2 + \ldots + a_nx_n = b\]where \(a_i\) and \(b\) are constants.

• In \(\mathbb{A}^{n}\), hyperplanes split the space into two parts, demonstrating their importance in both algebraic and geometric contexts.
• For \(\mathbb{A}^{2}\), this is simply a line, elucidating the equivalence between lines and hyperplanes in 2D space.

Understanding hyperplanes enables us to comprehend complex problems involving space division and solutions to linear equations across multiple dimensions.