Page 270 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes This is straightforward computation in which one uses the fact that ( | ) = 0 for j . k
j k
In the special case in which { . . . , } is an orthonormal set, Bessel’s inequality says that
1 n
2 2
( | ) .
k
k
The corollary also tells us in this case that is in the subspace spanned by , . . ., if and only
1 n
if

= ( | )
k k
k
or if and only if Bessel’s inequality is actually an equality. Of course, in the event that V is finite
dimensional and { , . . . , } is an orthogonal basis for V, the above formula holds for every
1 n
vector in V. In other words, if { , . . . , } is an orthonormal basis for V, the kth coordinate of
1 n
in the ordered basis { , . . . , } is ( | ).
1 n k
Example 15: We shall apply the last corollary to the orthogonal sets described in Example
11. We find that

n 1 2 1
2
(a) f ( )e –2 ikt dt f ( ) dt
t
t
0 0
k n
2
1 n n
(b) c e 2 ikt dt c 2
k k
0
k n k n
1
2
(c) 2 cos 2 t 2 sin 4 t dt 1 1 2.
0
Self Assessment

3. Apply the Gram-Schmidt process to the vectors = (1, 0, 1), (1, 0, –1), = (0, 3, 4) to
1 2 3
3
obtain an orthonormal basis for R with the standard inner product.
4. Let V be an inner product space. The distance between two vectors and in V is defined
by

d ( , ) ,
so that
(a) d( , ) 0;

(b) d ( , ) = d ( , );
(c) d( , ) d( , ) + ( , ).

24.3 Summary

 The idea of an inner product is somewhat similar to the scalar product in the vector
calculus.

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