Linear Transformations and Matrices

A linear transformation $T: V \to W$ is a mapping between vector spaces that preserves the operations of addition and scalar multiplication.

Properties of Linear Maps

$T$ is linear if for all $\mathbf{u}, \mathbf{v} \in V$ and $c \in F$:

1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. $T(c\mathbf{v}) = cT(\mathbf{v})$

Every linear transformation between finite-dimensional vector spaces can be represented as a matrix.

import numpy as np

def apply_transformation(A, v):
  return np.dot(A, v)

# Rotation Matrix by 90 degrees (pi/2)
theta = np.pi / 2
R = np.array([[np.cos(theta), -np.sin(theta)],
            [np.sin(theta),  np.cos(theta)]])

v = np.array([1, 0])
v_rot = apply_transformation(R, v)

print(f"Original: {v}")
print(f"Rotated 90 deg: {v_rot.round(2)}")

Composition and Matrix Multiplication

If $T: U \to V$ and $S: V \to W$ are linear maps, their composition $S \circ T: U \to W$ is also linear. The matrix representing $S \circ T$ is the product of the matrices representing $S$ and $T$.

Change of Basis

The matrix representation of a linear map depends on the choice of bases for $V$ and $W$. If $[T]_B$ is the matrix in basis $B$, and $P$ is the transition matrix from basis $B'$ to $B$, then: $[T]_{B'} = P^{-1} [T]_B P$

# T in standard basis
T_std = np.array([[1, 2], [3, 4]])

# New basis B' = {[1,1], [1,0]}
P = np.array([[1, 1], [1, 0]])
P_inv = np.linalg.inv(P)

# T in new basis
T_new = P_inv @ T_std @ P
print(f"Matrix in new basis:\n{T_new.round(2)}")