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Linear Transformations

Definition

Prereq knowledge: Basis Vectors

Matrices are essentially transformation of space. 

Lineaer Transformations abide by 2 rules

with the grey grid being the input, and purple grid as the output. 

Screenshot from 2024-02-02 21-50-08.png

 

Matrices can encode geometric operations such as rotation, reflection and transformation. Thus if a collection of vectors represents the vertices of a three-dimensional geometric model in Computer Aided Design software then multiplying these vectors individually by a pre-defined rotation matrix will output new vectors that represent the locations of the rotated vertices. This is the basis of modern 3D computer graphics.

Matrix-Matrix Multiplication

Transformations

Rotate

Shear 

Scaling

 

Matrix Addition

Matrix-Scalar Addition 

test

Matrix-Matrix Addition

Matrix Multiplication

Matrix Transpose


Scalar-Matrix Multiplication

Hadamard Product