Subspaces and Quotients

A vector space can contain smaller vector spaces called subspaces. We can also “divide” a space by a subspace to create a quotient space.

Subspaces

A subset $W$ of a vector space $V$ is a subspace if:

1. The zero vector $\mathbf{0} \in W$.
2. $W$ is closed under addition: $\mathbf{u}, \mathbf{v} \in W \implies \mathbf{u} + \mathbf{v} \in W$.
3. $W$ is closed under scalar multiplication: $c \in F, \mathbf{v} \in W \implies c\mathbf{v} \in W$.

import numpy as np

def is_in_subspace(vector, basis):
  """Check if a vector is in the subspace spanned by a basis."""
  # Solve basis * c = vector
  A = np.array(basis).T
  try:
      x, residuals, rank, s = np.linalg.lstsq(A, vector, rcond=None)
      # If residuals are near zero, it's in the subspace
      return np.allclose(np.dot(A, x), vector)
  except:
      return False

subspace_basis = [[1, 0, 0], [0, 1, 0]]
v1 = [0.5, 2.0, 0.0]
v2 = [0.5, 2.0, 1.0]

print(f"Vector {v1} in subspace: {is_in_subspace(v1, subspace_basis)}")
print(f"Vector {v2} in subspace: {is_in_subspace(v2, subspace_basis)}")

Quotient Spaces

Given a subspace $W \subseteq V$, the quotient space $V/W$ is the set of all cosets $\mathbf{v} + W$. Intuitively, the quotient space “ignores” all differences that lie within $W$.

The dimension of a quotient space is: $\dim(V/W) = \dim(V) - \dim(W)$

Kernels and Images

For a linear map $T: V \to W$:

• The Kernel $\ker(T) = \{ \mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0} \}$ is a subspace of $V$.
• The Image $im(T) = \{ T(\mathbf{v}) \mid \mathbf{v} \in V \}$ is a subspace of $W$.

import numpy as np
from scipy.linalg import null_space

A = np.array([[1, 2, 3], [2, 4, 6], [1, 1, 1]])

# Find basis for kernel
ns = null_space(A)
print(f"Basis for Kernel (Null Space):\n{ns}")

# Find rank (dimension of image)
rank = np.linalg.matrix_rank(A)
print(f"Rank (dim of Image): {rank}")