# gram-schmidt procedure

• Nov 7th 2010, 11:32 AM
zebra2147
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 $\displaystyle P_{2}(R)$.
• Nov 7th 2010, 12:09 PM
HallsofIvy
Quote:

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"?
• Nov 7th 2010, 01:33 PM
zebra2147
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.