Basis and Dimension

The concepts of basis and dimension provide a way to “measure” the size and complexity of a vector space.

Linear Independence and Spanning

A set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is linearly independent if the only solution to $c_1\mathbf{v}_1 + \dots + c_n\mathbf{v}_n = \mathbf{0}$ is $c_i = 0$ for all $i$ . The span of a set is the set of all possible linear combinations.

Interactive Lab

import numpy as np

def check_linear_independence(vectors):
  matrix = np.array(vectors).T
  rank = np.linalg.matrix_rank(matrix)
  is_independent = rank == len(vectors)
  return is_independent, rank

# Test linearly independent vectors
v1 = [1, 0, 0]
v2 = [0, 1, 0]
print(f"Independent: {check_linear_independence([v1, v2])}")

# Test linearly dependent vectors
v3 = [1, 1, 0]
print(f"Independent with v3: {check_linear_independence([v1, v2, v3])}")

python

1import numpy as np

3def check_linear_independence(vectors):

4 matrix = np.array(vectors).T

5 rank = np.linalg.matrix_rank(matrix)

6 is_independent = rank == len(vectors)

7 return is_independent, rank

9# Test linearly independent vectors

10v1 = [1, 0, 0]

11v2 = [0, 1, 0]

12print(f"Independent: {check_linear_independence([v1, v2])}")

14# Test linearly dependent vectors

15v3 = [1, 1, 0]

16print(f"Independent with v3: {check_linear_independence([v1, v2, v3])}")

Definition of a Basis

A basis $B$ for a vector space $V$ is a set of vectors that:

Is linearly independent.
Spans $V$ .

How many vectors are in any basis of R^3?

Dimension

The dimension of a vector space $V$ , denoted $\dim(V)$ , is the number of vectors in any basis for $V$ .

Interactive Lab

import numpy as np

def get_dimension(matrix):
  # The dimension of the column space is the rank
  return np.linalg.matrix_rank(matrix)

A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(f"Rank (Dimension of Col Space) of A: {get_dimension(A)}")
print("A rank of 2 means the vectors are coplanar in R^3.")

python

1import numpy as np

3def get_dimension(matrix):

4 # The dimension of the column space is the rank

5 return np.linalg.matrix_rank(matrix)

7A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

8print(f"Rank (Dimension of Col Space) of A: {get_dimension(A)}")

9print("A rank of 2 means the vectors are coplanar in R^3.")

Basis and Dimension

Basis and Dimension

Linear Independence and Spanning

Definition of a Basis

How many vectors are in any basis of R^3?

Dimension

If a subspace of R^5 has dimension 0, what is in the subspace?