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Visualizing Linear Transformations

  1. What we aim to do here

Linear transfors are one of the core concepts in linear algebra. A large number of concepts in linear algebra can be understood with a great understanding of the linear transforms. Through this project, we aim to create a tool that would help in the visualization of linear algebra concepts through linear transformations in the 2D coordinate plane.

  1. Project design

We aim to design a 2D cartesian plane with the help of pygame library. Features that we aim to give:

  1. Creating a 2D plane :

https://github.com/DeVcB13d/Linear-Transform-Visualisation/tree/main/media/pg1.jpg

  • The above shown picture is a model of the 2D plane that we aim to implement
  • Critical features to be added
    • Basic coordinate plane with number markings and stuff
    • Getting back the coordinate in cartesian system and the vice versa.
    • Need to able to move the plane with keys and the number markings should change appropriately
    • To be able to zoom into the plane
    • Also the plane should be able to move properly
    • Add the linear transform change to the plane (blue lines)
  • Creating an interactive interface using tkinter : 1.Making two frames :To enter two vectors for transformation 2.Creating two buttons:Run simulation and calculating transformed vector.
  1. Adding linear algebra features to the tool
  • Once the linear tranformation part is clearly done and visualized the next step is to add some linear algebra functionalities
    1. Solving a linear equation : The solution to a linear equation AX = b is the coordinate of the point that would eventually reach b after the transform A
    2. Matrix multiplications and its applications : Matrix multiplication AXB is applying a transformation on the A on the standard plane and applying B on the new transformed plane.
    3. Null Space of a matrix : The vectors that go into 0 vector after the transformation A.
    4. Eigenvectors : These vectors would only get streched by a certain amount on applying the transformation, so we have to highlight the vectors that would only get streched
    5. Finding the determinant : change in area after transformation.

Design implementaion

  1. PHASE 1 : Designing the 2D plane:

Desired Properties:

  • Need to be extremely generalized and be open for expansion
  • Should be able to apply the linear algebra operations on the plane easily

Ideas/Algorithms for design:

  1. Designing a mathematically accurate 2D plane with various features :

Ideas to add Linear algebra functionalities