Next:
Orthonormal bases.
Up:
Euclidean spaces.
Previous:
Verification of the axioms
Orthogonality.
Definition 12.3.1
Let
be an
vector space with an inner product. Two vectors
and
are
orthogonal
if their inner product
is equal to 0.
Example 12.3.2
Let
with the ``regular'' inner product defined in Example
1.3
.
Take
and
. Then we have:
, i.e. the vectors
and
are orthogonal.
Take
and
. Then we have:
, i.e. the vectors
and
are not orthogonal.
The vector of the standard basis of
are orthogonal.
Example 12.3.3
Let
be the space of all the polynomials in one variable
x
with coefficients in
, with inner product:
If
f
_{1}
(
x
)=
x
and
f
_{2}
(
x
)=
x
^{2}
, then
. Thus the functions
f
_{1}
and
f
_{2}
are not orthogonal vectors in
.
Example 12.3.4
Let
, with the inner product
Then the sine and the cosine function are orthogonal.
Noah Dana-Picard
2001-02-26