Page 274 - 35Linear Algebra

274 Orthonormal Bases and Complements

from the solution set of the matrix equation

T
v x = 0 .
1
Pick the vector v 2 obtained from the standard Gaussian elimina-
tion procedure which is the coefficient of x 2 .
(b) Pick a vector perpendicular to both v 1 and v 2 from the solutions
set of the matrix equation
T
v 1 x = 0 .
v T
2
Pick the vector v 3 obtained from the standard Gaussian elimina-
tion procedure with x 3 as the coefficient.
(c) Pick a vector perpendicular to v 1 , v 2 , and v 3 from the solution set
of the matrix equation
 
v 1 T
T
v  x = 0 .


2
v 3 T
Pick the vector v 4 obtained from the standard Gaussian elimina-
tion procedure with x 3 as the coefficient.
(d) Normalize the four vectors obtained above.

5. Use the inner product
1
Z
f · g := f(x)g(x)dx
0
2
3
on the vector space V = span{1, x, x , x } to perform the Gram-Schmidt
2
3
procedure on the set of vectors {1, x, x , x }.
6. Use the inner product
Z 2π
f · g := f(x)g(x)dx
0
on the vector space V = span{sin(x), sin(2x), sin(3x)} to perform the
Gram-Schmidt procedure on the set of vectors {sin(x), sin(2x), sin(3x)}.
Try to build an orthonormal basis for the vector space

span{sin(nx) | n ∈ N} .

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