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Can somebody explain me the relation between the orthogonal complement of a vector and the orthonormal basis of a vector?

Simply,how can we find the orthogonal complement of a row/column vector using Gram-Schmidt algorithm?

If S is the linear span of the column vectors a=(1,4,1) and b=(-1,0,1).How can we find the orthogonal complement of S using Gram-Schmidt algorithm?

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    Perhaps by "orthogonal complement of a vector" you mean the orthogonal complement of the vector space spanned by a vector, but "a basis of a vector", let alone orthonormal, is more than I can parse.2012-12-05

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I don't think you need to use G-S to find the orthogonal complement of a vector space. In this case, since $\,\dim S=2\,$, we get that $\,\dim S^\perp=1\,$ and assuming you mean the usual, euclidean inner product in $\,\Bbb R^3\,$:

$(x,y,z)\perp (1,4,1)\Longrightarrow x+4y+z=0$

$(x,y,z)\perp (-1,0,1)\Longrightarrow x+z=0$

Comparing both equations above we get the conditions $\,y=0\,,\,z=-y\,$, and thus $\,S^\perp=...\,$