Examples

1. System of linear equations

   from matrixflow import solve_linear_system
   
   print(
       solve_linear_system(
           [
               [2, 3],
               [5, -6],
           ],
           [4, -4],
       ) # Solves linear system using matrix inverse operations
   )
   
   # Alternatively, this solves it using gaussian elimination
   print(
       Matrix([
           [2, 3],
           [5, -6],
       ])
       .reduced_row_echelon_form(
           Matrix.from_columns([[4, -4]])
       )[1]
       .col_at(0)
   ) # this prints the same thing as the above print statement
   

   This is mathematically equivalent to:

   \(\begin{cases}2x+3y=4\\5x-6y=-4\end{cases}\)

   \(\therefore x=\frac{4}{9}, y=\frac{28}{27}\)

2. Linear transformations

   from matrixflow import Matrix, Vector
   
   v = Vector([1, 2])
   shear = Matrix.shear_2d(2)
   reflection = Matrix.reflect_x()
   
   # shear -> reflect across x-axis
   print(reflection @ shear @ v)
   
   # Rank: shear does not reduce space to a lower dimension: remains 2D
   print(shear.rank()) # 2
   
   # this non-square matrix linear transformation increases the dimension of space from R^2 -> R^3
   print(
       Matrix([
           [1, 2],
           [4, 5],
           [7, 8],
       ]) @ v
   )