What is the orthonormalization of √3(x^2 – 1), √3(x^2 + x + 2)?
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with the inner product defined as follows.
p, q = a0b0 + a1b1 + a2b2
Use the Gram-Schmidt orthonormalization process to make this orthonormalized.
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The two vectors are
a = (sqrt(3) 0 -sqrt(3) ) = sqrt(3) (1 0 1)
and
b = (sqrt(3) sqrt(3) 2 sqrt(3)) = sqrt(3) (1 1 2)
In Gram-Schmidt orthogonalisation you first substract from a the component of a in the direction of b, to obtain an a' that is by construction orthogonal to b:
a'= a - (a dot b) b / |b|^2
Now, a dot b = 3 * (1 * 1 + 0 * 1 + 1 * 2) = 9 and |b|^2 = 18
So a'= sqrt(3) ( 1 0 1) - 9 sqrt(3) ( 1 1 2) /18= 1/2 sqrt(3) ( 1 -1 0)
All that is left is to normailise a' and b
|a'| = sqrt( 3/4 (1^2 + (-1)^2 + 0^2) ) = sqrt(3/2)
|b| = 3 sqrt(2)
Hence the orthonormal vectors are
1/sqrt(2) (1 -1 0)) and 1/sqrt(6) ( 1 1 2)
(x3 + 4x2 - 2x + 5) - (3x3 - 3x2 + x - 2) Distribute the -a million to the 2d section: x^3 + 4x^2 - 2x + 5 - 3x^3 + 3x^2 - x + 2 combine like words: -2x^3 + 7x^2 - 3x + 7 answer S.