Linear Transformations
Linear Transformation of Matrices
Prereq knowledge: Vectors and Basis Vectors, Linear dependent and independent
Linear Transformation is a way to move around space such that it fulfills these two conditions:
- All grid lines must remain lines
- Origin (0,0) must remain fixed
Transformations (functions f(x) - best described as a movement from input to its output)
The following are Linear Transformation where the grey grid is the original:
The following are Non Linear Transformations
In 3D
With the above two conditions are fulfilled, we can deduce where ANY vector land as long as we have record of where i hat and j hat lands. In 2D space, this only requires two vectors
Example of i hat and j hat moving to their new space based on the formula (linear combination?)
A 2x2 Matrix is created through two vectors
The columns in the matrix are transformed versions of the basis vectors, and the result is the linear combination of those vectors