Let $f(x)=−3, g(x)=−9x+6$ and $h(x)=3x^{2}, \langle p,q \rangle =\int_0^1 p(x)q(x)dx$

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace spanned by the functions $f(x), g(x)$, and $h(x)$. I've obtained the first base to be $-1$ but got the 2nd base to be $-9x+\frac{9}{2}$ But this was incorrect? Any help on the 2nd and 3rd base?

  • $\begingroup$ Please check if I formatted the 2nd base you got right. Did you mean $\frac{-9x + 9}{2}$ or $-9x + \frac92$? $\endgroup$ – RGS Nov 24 '16 at 18:15
  • $\begingroup$ Yes you formatted it correctly..Thanks!! $\endgroup$ – Pkr96 Nov 24 '16 at 18:19
  • 2
    $\begingroup$ You found an orthogonal basis, whereas the question was to find an orthonormal one $\endgroup$ – Qwerty Nov 24 '16 at 18:22
  • $\begingroup$ If you multiply the functions and apply the framework-Schmidt procedure, it must be true but then check how you are determining the inner product of the two chosen functions say <for,g> $\endgroup$ – DOCTOR NGILAZI BANDA JOSHUA Nov 24 '16 at 18:23
  • $\begingroup$ @Qwerty Do i have to normalise my base in that case? $\endgroup$ – Pkr96 Nov 24 '16 at 18:24

To find an orthonormal basis, the $2$nd base would be $${-9x+9/2\over||-9x+9/2|| }$$ where $$||-9x+9/2||^2 =\langle-9x+9/2,-9x+9/2\rangle=\int_0^1(-9x+9/2)^2dx$$


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