Linear Transformations
Linear transformations preserve vector addition and scalar multiplication. Every finite-dimensional linear map can be represented by a matrix, making algebra and geometry computable.
Key Points
- Linear maps satisfy $T(u+v) = T(u) + T(v)$ and $T(cu) = cT(u)$.
- The rank-nullity theorem relates dimensions of kernel and image.
- Change of basis expresses the same transformation in different coordinate systems.
Formulas
Linearity
$$T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)$$
Matrix representation
$$[T(v)]_B = [T]_B [v]_B$$
Rank-nullity
$$\operatorname{rank}(T) + \operatorname{nullity}(T) = \dim(V)$$
Code Example
import numpy as np
T = np.array([[1, 2], [3, 4], [5, 6]])
print('rank:', np.linalg.matrix_rank(T))
print('nullity:', T.shape[1] - np.linalg.matrix_rank(T))