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Math Help - LinearAlgebra- Gram-Schmidt Algorithm and linear combinations

  1. #1
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    LinearAlgebra- Gram-Schmidt Algorithm and linear combinations

    Hey guys, I have a problem relating to the Gram-Schmidt algorithm and expressing a vector as a linear combination of other vectors.

    LinearAlgebra- Gram-Schmidt Algorithm and linear combinations-5q1.jpg

    Here is my attempt at doing the question.

    LinearAlgebra- Gram-Schmidt Algorithm and linear combinations-5q1-attempta.pngA
    LinearAlgebra- Gram-Schmidt Algorithm and linear combinations-5q1-attemptb.jpgB

    So I would like to know if I have done this correctly, and if not, how do I go about solving this question?

    Help would be much appreciated.
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  2. #2
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    Re: LinearAlgebra- Gram-Schmidt Algorithm and linear combinations

    At the end, you have (1, 0, 1), (0, 2, 0), and (-1, 0, 1) as orthogonal vectors and then divide the vectors by \sqrt{2}, 2, and \sqrt{2} to get unit vectors. That is correct even with this non-standard inner product. The last part asks you to write (2, 1, -1) as a linear combination of these basis vectors. That means you want to find numbers, a, b, and c, such that a(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})+ b(0, 1, 0)+ c(-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})= (2, 1, -1).

    You could do this by solving the three equations \frac{a}{\sqrt{2}}- \frac{c}{\sqrt{2}}= 2, b= 1, \frac{a}{\sqrt{2}}+ \frac{c}{\sqrt{2}}= 1

    But because this is an "orthonormal basis" there is a simpler way: take the dot product of [tex](\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}) with both sides of the equation. That will give a= (\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})\cdot (2, 1, -1)= \frac{2}{\sqrt{2}}- \frac{1}{\sqrt{2}}= \frac{1}{\sqrt{2}}

    Take the dot product of (0, 1, 0) with both sides of the equation. That will give b= (0, 1, 0)\cdot (2, 1, -1)= 1

    Finally, take the dot product of (-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}) with both side of the equation to get c= (-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})\cdot (2, 1, -1)= -\frac{2}{\sqrt{2}}- \frac{1}{\sqrt{2}}= -\frac{3}{\sqrt{2}}.
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