6—Vector Spaces 1556.7 Bases and Scalar Products
When there is a scalar product, a most useful type of basis is the orthonormal one, satisfying
〈
~vi,~vj〉
=δij={
1 ifi=j
0 ifi 6 =j(11)
The notationδijrepresents the very useful Kronecker delta symbol.
In the example of Eq. ( 1 ) the basis vectors are orthonormal with respect to the scalar product in Eq. ( 6 ).
It is orthogonal with respect to the other scalar product mentioned there, but it is not in that case normalized to
magnitude one.
To see how the choice of even an orthonormal basis depends on the scalar product, a different scalar product
can be used on this space. Take the special case of two dimensions. The vectors are now pairs of numbers. Think
of the vectors as 2 × 1 matrix column and use the 2 × 2 matrix
(
2 1
1 2
)
The scalar product of two vectors is
〈
(a 1 ,a 2 ),(b 1 ,b 2 )〉
=
(a* 1 a* 2 )(
2 1
1 2
)(
b 1
b 2)
= 2a* 1 b 1 +a* 1 b 2 +a* 2 b 1 + 2a* 2 b 2 (12)To show that this satisfies all the defined requirements for a scalar product takes a small amount of labor. The
vectors that you may expect to be orthogonal,(1 0)and(0 1), are not.
In example 6, if we let the domain of the functions be−L < x <+Land the scalar product is as in
Eq. ( 8 ), then the set of trigonometric functions can be used as a basis.
sinnπx
Land cosmπx
L
n= 1, 2 , 3 ,... and m= 0, 1 , 2 , 3 ,....That a function can be written as a series
f(x) =∑∞
1ansinnπx
L+
∑∞
0bmcosmπx
L