Linear transformation: Find the necessary quantity of T

In summary, the conversation discusses a linear transformation and a problem involving the transformation of a disk with radius 1. The transformation is defined as T: (x,y) → (5x+3y)/4, (3x+5y)/4 and the goal is to find the lowest number of steps needed for T to contain at least 2019 points with integer coordinates. The conversation also explores the concept of changing basis and transforming a circle into an ellipse. The main question is whether to calculate the volume or area of the new shape.
  • #1
LCSphysicist
646
161
Homework Statement
.
Relevant Equations
..
> Let ##C## be the disk with radius 1 with center at the origin in ##R^2##.
> Consider the following linear transformation: ##T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})##
>
> What is the lowest number such that ##T^{n}(C)## contains at lest ##2019## points ##(a,b)##, with a and b integers.So we have ##x²+y² = 1##,

##T: (x,y)\to ( x + y + \frac{x-y}{4}, x + y - \frac{x-y}{4})##

##T²: (x',y') \to ( x + y + \frac{3(x-y)}{8}, x + y - \frac{3(x-y)}{8})##

To be pretty honest, i couldn't see any pattern that simplify the solution of this problem...
 
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  • #2
You should find the matrix of ##T## and look how the volume of the disc changes, or the square of length two around the origin to start with an easier area.
 
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  • #3
fresh_42 said:
You should find the matrix of ##T## and look how the volume of the disc changes.
$$
\begin{pmatrix}
x'\\ y'

\end{pmatrix}
=
\begin{pmatrix}
5/4 &3/4 \\
3/4 &5/4
\end{pmatrix}^{n}
\begin{pmatrix}
x\\ y

\end{pmatrix}$$

Apparently this changing of basis transform the circle (disc) in an ellipse. I think what you mean is to calculated the area of the new disc (ellipse)? (and not the volume?)
 
  • #4
LCSphysicist said:
$$
\begin{pmatrix}
x'\\ y'

\end{pmatrix}
=
\begin{pmatrix}
5/4 &3/4 \\
3/4 &5/4
\end{pmatrix}^{n}
\begin{pmatrix}
x\\ y

\end{pmatrix}$$

Apparently this changing of basis transform the circle (disc) in an ellipse. I think what you mean is to calculated the area of the new disc (ellipse)? (and not the volume?)
Yes, volume is the general term for any dimension. Choose the unit square and see how the area will change with every step. What do you observe?
 
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Related to Linear transformation: Find the necessary quantity of T

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another vector space in a way that preserves the linear structure of the original space.

What are the necessary quantities needed to find a linear transformation?

The necessary quantities needed to find a linear transformation include: the original vector space, the target vector space, and a transformation matrix or equation.

How do you find the transformation matrix for a linear transformation?

To find the transformation matrix for a linear transformation, you first need to determine the basis vectors for the original and target vector spaces. Then, the transformation matrix can be constructed using the coefficients of the basis vectors in the transformation equation.

What is the purpose of finding the necessary quantity of T in a linear transformation?

The purpose of finding the necessary quantity of T in a linear transformation is to accurately map the original vector space to the target vector space, while preserving the linear structure of the original space.

Can a linear transformation have multiple necessary quantities?

Yes, a linear transformation can have multiple necessary quantities, as there are different ways to represent a transformation and different ways to map one vector space to another. However, the necessary quantities should all lead to the same transformation result.

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