- #1

Nusc

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In what context do they apply to?

How important is it that we treat them differently?

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- Thread starter Nusc
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- #1

Nusc

- 760

- 2

In what context do they apply to?

How important is it that we treat them differently?

- #2

lurflurf

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When speaking of two vectors u,v perpendicular and orthoganal are used interchangably to mean that an inner product is zero.Nusc said:

In what context do they apply to?

How important is it that we treat them differently?

<u|v>=0.

Perpendicular sometimes, but not always is used to indicate that the inner product in question has geometric interpitations. In that context <u|v> would mean two lines related to the vectors form right angles.

Orthoganal is applied to linearly independent sets to mean that for any two vectors in a set <v(i)|v(j)>=0 if i and j are not the same. Orthonormal means that in addition to being orthoganal <v(i)|v(i)>=1. This is quite use full because the problem of determining the coefficient of a vector in a representation of a vector by a basis is in general dependent on solving a linear system, but reduces in a orthoganal basis to finding an inner product.

because

v=a1v1+a2vi+...

so

<v(i)|v>=a(i)<v(i)|v(i)>

since the other terms are zero.

- #3

quasar987

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And the notion of orthogonality goes beyond that of vectors. For exemple, two functions f(x) and g(x) are said to be orthogonal over the interval [a,b] with weighting function w(x) if their inner product, defined as the integral of fgw from a to b, is 0. We also defined orthonormality between functions as "f(x) and g(x) are orthonormal over the interval [a,b] with weighting function w(x) if the integral of fgw from a to b, is 0 if f and g are not equal and is 1 if they are equal."

Note that you could have learned much of that by browsing on http://mathworld.wolfram.com/

- #4

Galileo

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Orthonormal means both orthogonal and normalized.

- #5

mathwonk

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