# example of Gram-Schmidt orthogonalization

Let us work with the standard inner product on ${\mathbb{R}}^{3}$ (dot product^{}) so we can get a nice geometrical visualization.

Consider the three vectors

${v}_{1}$ | $=(3,0,4)$ | ||

${v}_{2}$ | $=(-6,-4,1)$ | ||

${v}_{3}$ | $=(5,0,-3)$ |

which are linearly independent^{} (the determinant^{} of the matrix $A=({v}_{1}|{v}_{2}|{v}_{3})=116\ne 0)$ but are not orthogonal^{}.

We will now apply Gram-Schmidt to get three vectors ${w}_{1},{w}_{2},{w}_{3}$ which span the same subspace^{} (in this case, all ${R}^{3}$) and orthogonal to each other.

First we take ${w}_{1}={v}_{1}=(3,0,4)$. Now,

$${w}_{2}={v}_{2}-\frac{{w}_{1}\cdot {v}_{2}}{{\parallel {w}_{1}\parallel}^{2}}{w}_{1}$$ |

that is,

$${w}_{2}=(\frac{-108}{25},-4,\frac{81}{25})$$ |

and finally

$${w}_{3}={v}_{3}-\frac{{w}_{1}\cdot {v}_{3}}{{\parallel {w}_{1}\parallel}^{2}}{w}_{1}-\frac{{w}_{2}\cdot {v}_{3}}{{\parallel {w}_{2}\parallel}^{2}}{w}_{2}$$ |

which gives

$${w}_{3}=(\frac{1856}{1129},\frac{3132}{1129},\frac{1392}{1129})$$ |

and so $\{{w}_{1},{w}_{2},{w}_{3}\}$ is an orthogonal set of vectors such that $\u27e8{w}_{1},{w}_{2},{w}_{3}\u27e9=\u27e8{v}_{1},{v}_{2},{v}_{3}\u27e9$.

If we rather consider $\{{w}_{1}/\parallel {w}_{1}\parallel ,{w}_{2}/\parallel {w}_{2}\parallel ,{w}_{3}/\parallel {w}_{3}\parallel \}$ then we get an orthonormal set^{}.

Title | example of Gram-Schmidt orthogonalization^{} |
---|---|

Canonical name | ExampleOfGramSchmidtOrthogonalization |

Date of creation | 2013-03-22 15:03:02 |

Last modified on | 2013-03-22 15:03:02 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 5 |

Author | drini (3) |

Entry type | Example |

Classification | msc 65F25 |

Related topic | ProofOfGramSchmidtOrthogonalizationProcedure |