I'm currently learning Linear Algebra and I was asked to calculate the orthogonal projection of the vector $x^3$ on a subspace of $R_5[x]$ - $U = Sp\{1, x, x^2\}$ with the integral dot product in $[0,1]$.
I'm trying to get the orthogonal base of $U$, but I'm having trouble with the Gram-Schmidt process.
Can someone please help me get the orthogonal base of $U$? I keep getting the same non-orthogonal base using the Gram-Schmidt process...
The base I get is $\{1, x-\frac12, x^2-x-\frac56\}$, and the dot product of the last two vectors isn't zero...
There's another problem with your answer: the second and the third functions don't have norm $1$. Did you forget to divide by the norm? What I got was$$\bigl\{1,\sqrt3(2x-1),\sqrt5\left(6x^2-6x+1\right)\bigr\}.$$