1. ## gram-schmidt procedure

I could use some help with the following...

On $\displaystyle P_{2}(R)$, consider the inner product given by $\displaystyle <p,q>=\int_0^1 p(x)q(x) dx$. Apply the Gram-Schmidt procedure to the basis $\displaystyle (1,x,x^2)$ to produce an orthonormal basis of $\displaystyle P_{2}(R)$.

2. Originally Posted by zebra2147
I could use some help with the following...

On $\displaystyle P_{2}(R)$, consider the inner product given by $\displaystyle <p,q>=\int_0^1 p(x)q(x) dx$. Apply the Gram-Schmidt procedure to the basis $\displaystyle (1,x,x^2)$ to produce an orthonormal basis of $\displaystyle P_{2}(R)$.
Okay, well, what do you understand as the"Gram-Schmidt procedure"? To start with you will want unit vectors. What is the length of the vector "1"? What is the inner product of "1" and "x"?

3. Well, the way I understand it is that we are trying to find an orthonormal list of vectors in V. Here is the procedure that I believe is accurate...
$\displaystyle e_1=v_1/||v_1||$
$\displaystyle e_2=v_{2}-<v_{2},e_{1}>e_{1}$ where $\displaystyle <v_2,e_1>e_1$ is the projection of $\displaystyle v_2$ onto $\displaystyle e_1$. This process continues for all $\displaystyle e_j$.
The length of vector 1 is 1.
And the inner product of 1 and x is just x I believe.