File:Gram-Schmidt orthonormalization process.ogv

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Original file ‎(Ogg Theora video file, length 1 min 8 s, 1,280 × 720 pixels, 661 kbps, file size: 5.34 MB)

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English: The animation illustrates the Gram-Schmidt process for obtaining an orthonormal basis of vectors, in this particular case, in 3-dimensional euclidean space.
  1. The first of the original vectors, v1, is taken as the first vector of the orthogonal basis, u1.
  2. 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
  3. 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.
  4. 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.
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Author Lucas V. Barbosa
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Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
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current04:24, 2 February 20131 min 8 s, 1,280 × 720 (5.34 MB)LucasVB (talk | contribs){{Information |Description= {{en|1=The animation illustrates the Gram-Schmidt process for obtaining an orthonormal basis of vectors, in this particular case, in 3-dimensional euclidean space. # The fi...

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VP9 720P 94 kbps Completed 02:17, 28 August 2018 19 s
Streaming 720p (VP9) 94 kbps Completed 12:57, 19 June 2024 2.0 s
VP9 480P 51 kbps Completed 02:17, 28 August 2018 17 s
Streaming 480p (VP9) 51 kbps Completed 02:41, 14 March 2024 1.0 s
VP9 360P 32 kbps Completed 02:17, 28 August 2018 12 s
Streaming 360p (VP9) 32 kbps Completed 12:57, 19 June 2024 0.0 s
VP9 240P 20 kbps Completed 02:16, 28 August 2018 11 s
Streaming 240p (VP9) 20 kbps Completed 08:57, 22 December 2023 4.0 s
WebM 360P 86 kbps Completed 04:27, 2 February 2013 15 s
Streaming 144p (MJPEG) 418 kbps Completed 08:03, 14 November 2023 2.0 s

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