Gram-Schmidt Calculator
Convert a set of vectors into an orthonormal basis using the Gram-Schmidt process.
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How Gram-Schmidt Process Works
The Gram-Schmidt process converts a set of linearly independent vectors into an orthonormal set:
- Start with the first vector: \( u_1 = v_1 \)
- For each subsequent vector \( v_i \):
- Project \( v_i \) onto the span of previous vectors
- Subtract the projection to get the orthogonal component
- Normalize to get the orthonormal vector
The formulas used are:
- Projection: \( \text{proj}_u(v) = \frac{\langle v, u \rangle}{\langle u, u \rangle}u \)
- Orthogonal component: \( u_i = v_i - \sum_{j=1}^{i-1} \text{proj}_{u_j}(v_i) \)
- Normalization: \( e_i = \frac{u_i}{\|u_i\|} \)
Properties of the resulting basis:
- All vectors are orthogonal to each other
- All vectors have unit length
- The span remains the same as the original set