Linear Algebra Difficulty: Intermediate

Vector Spaces

Vector spaces abstract the notion of vectors to any set satisfying linearity axioms. This abstraction enables unified treatment of functions, sequences, and tensors.

Key Points

  • A vector space is closed under addition and scalar multiplication.
  • Bases provide coordinate-independent representations; dimension is the size of any basis.
  • Subspaces, span, linear independence, and basis are foundational concepts.

Formulas

Subspace conditions
$$U \subseteq V \text{ is a subspace if } \mathbf{0} \in U, \text{ closed under addition and scalar multiplication.}$$
Dimension theorem
$$\dim(V) = \dim(\ker T) + \dim(\operatorname{im} T)$$

Code Example

import numpy as np

# Check if vectors span R^3 by rank
A = np.array([[1, 2, 0], [0, 1, 1], [1, 0, 1]])
print(np.linalg.matrix_rank(A))  # 3 => span R^3

Tags

  • abstraction
  • basis
  • dimension

References

  • Linear Algebra Done Right
    Sheldon Axler · Springer · source
  • Finite-Dimensional Vector Spaces
    Paul R. Halmos · Springer · source

Knowledge Graph