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Some geometry: Cauchy-Schwarz inequality.
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Euclidean spaces.
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Orthogonality.
Orthonormal bases.
Definition 12.4.1
Let
be a basis for the vector space
. If
, the basis is called an
orthogonal basis
for
.
Definition 12.4.2
An
orthonormal basis
of the vector space
is an orthogonal basis, all of whose vectors being unit vectors.
This means that in an orthonormal basis, the vectors are orthogonal and of length 1.
Example 12.4.3
Some orthonormal bases for
:
.
.
Please check them!
Example 12.4.4
Let
, with the inner product
Then the sine and the cosine function form an orthonormal basis of
.
Noah Dana-Picard
2001-02-26