Two-Dimensional Orthonormal Basis Generator

In linear algebra, an orthonormal basis is a set of vectors that are both mutually orthogonal (perpendicular to each other) and normalized (each has unit length). Constructing such a basis is fundamental for simplifying computations in vector spaces, as it allows any vector to be uniquely expressed as a linear combination of basis vectors using simple dot products.

The Gram-Schmidt Orthogonalization Process is a classical algorithm that transforms a set of linearly independent vectors into an orthonormal set spanning the same subspace. The algorithm works iteratively:

1. Normalize the first vector to obtain the first basis vector
2. For each subsequent vector:
   • Subtract projections onto all previously computed basis vectors (removing components parallel to existing basis vectors)
   • Check if the resulting vector has sufficient magnitude (greater than the tolerance threshold)
   • If yes, normalize the result to add a new orthonormal basis vector
   • If no, the vector was linearly dependent and is discarded

Mathematical Formulation:

Given input vectors v₁, v₂, ..., vₖ, the orthonormal basis vectors u₁, u₂, ..., uₙ are computed as:

$$u_1 = \frac{v_1}{|v_1|}$$

For each subsequent vector vᵢ:

$$w_i = v_i - \sum_{j=1}^{i-1} \text{proj}_{u_j}(v_i)$$

where the projection is: $\text{proj}_u(v) = (v \cdot u) \cdot u$

If $|w_i| > \text{tol}$, then: $u_i = \frac{w_i}{|w_i|}$

Your Task: Write a Python function that computes an orthonormal basis for the subspace spanned by a list of 2D vectors using the Gram-Schmidt process. The function should:

• Accept a list of 2D vectors and a tolerance value for determining linear independence
• Return a list of orthonormal vectors (unit length and mutually perpendicular) spanning the same subspace
• Discard any vector that becomes nearly zero after projection removal (norm ≤ tolerance)