Archivo: Gram-Schmidt orthonormalization process

Description: The animation illustrates the Gram-Schmidt process for obtaining an orthonormal basis of vectors, in this particular case, in 3-dimensional euclidean space. The first of the original vectors, v1, is taken as the first vector of the orthogonal basis, u1. The second of the original vectors, v2, is then projected onto u1. The projection is subtracted from v2, obtaining a new vector u2, which is orthogonal to u1 The third of the original vectors, v3, is projected onto u1. The result is subtracted from v3, obtained the intermediate vector v'3. This intermediate vector was used for illustration purposes, in order to show the orthogonalization against each individual vector one at a time. Next, this intermediate vector is projected onto u2, and the resulting vector is subtracted from v'3, resulting in the vector u3, which is orthogonal to both u1 and u2. At this point, we have an orthogonal basis, but not an orthonormal basis, as the vectors are not unit vectors. In the last step, the vectors are scaled by the inverse of their norms, a process called normalization. The resulting vectors, e1, e2 and e3 are mutually orthogonal, and are all unit vectors. Therefore, they form an orthonormal basis for the 3-dimensional euclidean space R3. This completes the process.
Title: Gram-Schmidt orthonormalization process
Credit: Own work
Author: Lucas V. Barbosa
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License: Public domain
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