Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
then Eq. (5.10) reduces to
hU
jWi ˆ X X
u*w i j ij ˆ X
u*
i
i
j
i
X j
!
w j ij ˆ X
u*w i i :
i
…5:11†
Note that Axiom B.2 implies that if a vector jUi is orthogonal to every vector
of the vector space, then jUi ˆ 0: since hUj
iˆ 0 for all jibelongs to the vector
space, so we have in particular h UU j i ˆ 0.
We will show shortly that we may construct an orthonormal basis from an
arbitrary basis using a technique known as the Gram±Schmidt orthogonalization
process.
Example 5.7
Let jUi ˆ…3 4i†j1i‡…5 6i†j2i and jWi ˆ…1 i†j1i‡…2 3i†j2i be two vectors
expanded in terms of an orthonormal basis j1i and j2i. Then we have, using
Eq. (5.10):
hUjUi ˆ…3 ‡ 4i†…3 4i†‡…5 ‡ 6i†…5 6i† ˆ86;
hWjWi ˆ…1 ‡ i†…1 i†‡…2 ‡ 3i†…2 3i† ˆ15;
hU
jWi ˆ…3 ‡ 4i†…1 i†‡…5 ‡ 6i†…2 3i† ˆ35 2i ˆ hWjUi*:
Example 5.8
If ~A and ~B are two matrices, where
!
!
~A ˆ a11 a 12
; ~B ˆ b11 b 12
;
a 21 a 22 b 21 b 22
then the following formula de®nes an inner product on M 22 :
~A
B ~ ˆ a 11 b 11 ‡ a 12 b 12 ‡ a 21 b 21 ‡ a 22 b 22 :
To see this, let us ®rst expand ~A and ~B in terms of the following base vectors
jiˆ 1 1 0
; j2i ˆ 0 1
; j3i ˆ 0 0
; j4i ˆ 0 0
;
0 0
0 0
1 0
0 1
~A ˆ a 11 j1i‡ a 12 ji‡ 2 a 21 j3i‡ a 22 j4i; ~B ˆ b 11 ji‡ 1 b 12 j2i‡ b 21 j3i‡ b 22 j4i:
The result follows easily from the de®ning formula (5.10).
Example 5.9
Consider the vector jUi, in a certain orthonormal basis, with components
!
1 ‡ i
p
‡ i
jUi ˆ
3
p
; i ˆ
1 :
208