Gram-Schmidt Process in Sage

The Gram-Schmidt process is an algorithm that takes a linearly independent set of vectors and produces an orthonormal set of vectors that span the same subspace. This process is a fundamental tool in linear algebra and has many applications in mathematics and other fields.

SageMath is a free open-source mathematics software system that provides a wide range of mathematical tools, including the Gram-Schmidt process. In this tutorial, we will introduce the Gram-Schmidt process in Sage and show how it can be used to compute orthonormal bases for subspaces.

Theory

Suppose we have a linearly independent set of vectors $v_1, v_2, ..., v_n \in \mathbb{R}^m$. The Gram-Schmidt process constructs a set of orthonormal vectors $q_1, q_2, ..., q_n$ that span the same subspace as the original set of vectors.

The process can be performed iteratively as follows:

1. Set $q_1 = \frac{v_1}{||v_1||}$.
2. For each $i = 2, 3, ..., n$, compute $q_i$ as follows:
   • Compute $p_i = v_i - \sum_{j=1}^{i-1} \langle v_i, q_j \rangle q_j$.
   • Set $q_i = \frac{p_i}{||p_i||}$.

At each step, the vector $p_i$ is the projection of $v_i$ onto the subspace spanned by the orthogonal vectors $q_1, ..., q_{i-1}$.

Sage Implementation

In Sage, the Gram-Schmidt process can be implemented using the gram_schmidt() function. This function takes a list of vectors as input and returns a list of orthonormal vectors that span the same subspace.

Here is an example:

v = [vector([1, 2, 3]), vector([3, 2, 1]), vector([-1, 0, 2])]
q = gram_schmidt(v)

The input parameter v is a list of three vectors in $\mathbb{R}^3$. The output q is a list of three orthonormal vectors that span the same subspace as v. We can verify that the vectors in q are indeed orthonormal by computing their inner products:

for i in range(len(q)):
    for j in range(i, len(q)):
        print("q{} * q{} = {}".format(i+1, j+1, q[i].dot_product(q[j])))

This will output:

q1 * q1 = 1
q1 * q2 = 0
q1 * q3 = 0
q2 * q2 = 1
q2 * q3 = 0
q3 * q3 = 1

Conclusion

The Gram-Schmidt process is a powerful tool for constructing orthonormal bases for subspaces. In Sage, this process can be implemented using the gram_schmidt() function, which takes a list of vectors as input and returns a list of orthonormal vectors. This function is easy to use and can save time when computing orthonormal bases.