Linear independence

This blog is based on Jong-han Kim’s Linear Algebra

Linear dependence

set of $n$-vectors ${a_1, \dots, a_k} \ (\text{with } k \geq 1)$ is linearly dependent if

\[\beta_1 a_1 + \dots + \beta_k a_k = 0\]

holds for some $\beta_1, \dots, \beta_k$ that are not all zero

Linear independence

set of $n$-vectors ${a_1, \dots, a_k} \ (\text{with } k \geq 1)$ is linearly independent if

\[\beta_1 a_1 + \dots + \beta_k a_k = 0\]

holds only when $\beta_1 = \dots = \beta_k = 0$

e.g., the unit $n$-vectors $e_1, \dots, e_n$ are linearly independent

Linear combinations of linearly independent vectors

suppose $x$ is linear combination of linearly independent vectors $a_1, \dots, a_k$

\[x = \beta_1 a_1 + \dots + \beta_k a_k\]

the coefficients $\beta_1, \dots, \beta_k$ are unique, i.e. if

\[x = \gamma_1a_1 + \dots + \gamma_ka_k\]

then $\beta_i = \gamma_i$ for $i = 1, \dots, k$

this means that (in principle) we can deduce the coefficients from $x$
to see why, note that

\[(\beta_1 - \gamma_1)a_1 + \dots + (\beta_k - \gamma_k)a_k = 0\]

and so (by linear independence) $\beta_1 - \gamma_1 = \dots = \beta_k - \gamma_k = 0$

Independence-dimension inequality

  • a linearly independent set of $n$-vectors can have at most $n$ elements
  • put another way: any set of $n+1$ or more $n$-vectors is linearly dependent

Basis

a set of $n$ linearly independent $n$-vectors $a_1, \dots, a_n$ is called a basis
any $n$-vector $b$ can be expressed as a linear combination of them

\[b = \beta_1a_1 + \dots + \beta_n a_n\]

for some $\beta_1, \dots, \beta_n$

and these coefficients are unique
formula above is called expansion of $b$ in the $a_1, \dots, a_n$ basis

e.g., $e_1, \dots, e_n$ is a basis, expansion of $b$ is

\[b = b_1e_1 + \dots + b_ne_n\]

Orthonormal vectors

set of $n$-vectors $a_1, \dots, a_k$ are (mutually) orthogonal if $a_i \bot a_j$ for $i \neq j$
they are normalized if $\Vert a_i\Vert = 1$ for $i = 1, \dots, k$
they are orthonormal if both hold
can be expressed using inner products as

\[a^T_ia_j = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}\]

orthonormal sets of vectors are linearly independent
by independent-dimension inequality, must have $k \leq n$
when $k = n, a_1, \dots, a_n$ are an orthonormal basis

Orthonormal expansion

if $a_1, \dots, a_n$ is an orthonormal basis, we have for any $n$-vector $x$

\[x = (a^T_1 x)a_1 + \dots + (a^T_nx)a_n\]

called orthonormal expansion of $x$ (in the orthonormal basis)
to verify formula, take inner product of both sides with $a_i$

Gram-Schmidt (orthonormalization) algorithm

  • an alorithm to check if $a_1, \dots, a_k$ are linearly independent
  • we’ll see later it has many other uses
\[\begin{align*} & \textbf{given } \textit{n}\text{-vectors } a_1, \dots, a_k \\ & \textbf{for } i = 1, \dots, k \\ & \quad \text{1. Orthogonalization: } \tilde{q}_i = a_i - (q_1^T a_i)q_1 - \cdots - (q_{i-1}^T a_i)q_{i-1} \\ & \quad \text{2. Test for linear dependence: if } \tilde{q}_i = 0, \text{ quit} \\ & \quad \text{3. Normalization: } q_i = \tilde{q}_i / \| \tilde{q}_i \| \end{align*}\]

if G-S does not stop early (in step 2), $a_1, \dots, a_k$ are linearly independent
if G-S stops early in iteration $i=j$, then $a_j$ is a linear combination of $a_1, \dots, a_{j-1}$ (so $a_1, \dots, a_k$ are linearly dependent)

Code

import numpy as np

def gram_schmidt(vectors):
  A = np.array(vectors, dtype=float).T
  n, k = A.shape

  Q = np.zeros((n, k))

  for i in range(k):
    a_i = A[:, i]
    q_tilde = a_i

    for j in range(i):
      q_j = Q[:, j]
      projection = np.dot(q_j.T, a_i) * q_j
      q_tilde = q_tilde - projection

    norm_q_tilde = np.linalg.norm(q_tilde)

    if norm_q_tilde < 1e-10:
      print("linear dependent")
      return Q[:, :i]

    Q[:, i] = q_tilde / norm_q_tilde

  return Q

a1 = [1, 1, 0]
a2 = [2, 0, 1]
a3 = [0, 1, 2]
vectors_independent = [a1, a2, a3]

orthonormal_vectors = gram_schmidt(vectors_independent)
print(orthonormal_vectors.T)
print(np.dot(orthonormal_vectors.T, orthonormal_vectors))

print('='*30)

b1 = [1, 0, 1]
b2 = [0, 1, 1]
b3 = [2, 1, 3]
vectors_dependent = [b1, b2, b3]

orthonormal_vectors_dep = gram_schmidt(vectors_dependent)

print(orthonormal_vectors_dep.T)

[[ 0.70710678  0.70710678  0.        ]
 [ 0.57735027 -0.57735027  0.57735027]
 [-0.40824829  0.40824829  0.81649658]]
[[ 1.00000000e+00  2.50235355e-16  1.47380436e-17]
 [ 2.50235355e-16  1.00000000e+00 -3.33168241e-17]
 [ 1.47380436e-17 -3.33168241e-17  1.00000000e+00]]
==============================
linear dependent
[[ 0.70710678  0.          0.70710678]
 [-0.40824829  0.81649658  0.40824829]]