gram-schmidt procedure

**zebra2147** · November 7th 2010, 11:32 AM

I could use some help with the following...

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

**HallsofIvy** · November 7th 2010, 12:09 PM

Originally Posted by zebra2147

I could use some help with the following...

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

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"?

**zebra2147** · November 7th 2010, 01:33 PM

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...
$e_1=v_1/||v_1||$
$e_2=v_{2}-<v_{2},e_{1}>e_{1}$ where $<v_2,e_1>e_1$ is the projection of $v_2$ onto $e_1$ . This process continues for all $e_j$ .
The length of vector 1 is 1.
And the inner product of 1 and x is just x I believe.

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