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    Orthogonal and orthonormal basis

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    Let W be the subspace of V4(R) spanned by the following set of vectors...

    © BrainMass Inc. brainmass.com December 24, 2021, 7:17 pm ad1c9bdddf
    https://brainmass.com/math/geometry-and-topology/orthogonal-orthonormal-basis-176799

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    Let W be the subspace of V4(R) spanned by the following set of vectors:

    Use the Gram-Schmidt orthogonalisation process to find an orthogonal basis for W. Find also an orthonormal basis for W.

    The Gram-Schmidt Theorem:
    Given a basis for subspace W of Vn define:

    Then is an orthogonal basis of W and has the same span as

    We shall use the standard inner product:

    In our case:

    And:

    In the next step we have:

    Continued on next page

    Thus the orthogonal base is:

    The orthonormal basis is defined as the basis vectors divided by their length:

    We already know that:

    Therefore the orthonormal basis is:

    In column form:

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 7:17 pm ad1c9bdddf>
    https://brainmass.com/math/geometry-and-topology/orthogonal-orthonormal-basis-176799

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