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executor_2k2

Explanation of vector spaces and gram-schmidt orthogonalization

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executor_2k2    122
Does anyone get this? If you do, then can you explain it to me? I dont understand what the use of vector spaces is? I also dont understand what makes a vector orthogonal. Is it just perpendicular to the other directions? Thx

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Chock    122
The use of a vector space? That´s a good question. But I have no good answer

A vector can not be orthogonal. It can be orthogonal to another vector. Two vectors are orthogonal, if their scalar product is zero. So, (a,b,c) is orthogonal to (d,e,f) if a*d+b*e+c*f=0. This means, they form an angle of 90 degrees.

A matrix is orthogonal, if every two row (or column, does not matter) vectors are orthogonal.

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grhodes_at_work    1385
Chock's description of orthogonal is really a description of orthogonal vectors.

He is right to say that two vectors are orthogonal if their scalar product is zero---if they are perpendicular.

He is wrong about the definition of an orthogonal matrix. A matrix is orthogonal by definition only if a matrix times its transpose results in the identity matrix (diagonal elements = 1, off-diagonal elements = 0). Thus, a matrix can be orthogonal only if it its row vectors have length 1. The row vectors of an orthogonal matrix are called "orthonormal" since they are both orthogonal to each other and have length 1. I know its confusing. Think about it for a while.

See mathworld.wolfram.com for more info.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Edited by - grhodes_at_work on February 5, 2002 2:12:43 PM

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