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Question from my last LA.II assignment. I have no idea what to with it. It looked simple but now I think I don't even understand the question.

Consider the Inner Product Space P2 with the evaluation inner product at points x

Evaluation inner product points are defined such that <

First I have to find an orthogonal basis for W. But I can't use three basis vectors because the question demands a combination of two vectors. — I have no idea where to go. I used Gram-Smidt to come up with two orthogonal bases

And now I'm stuck. I assume evaluation inner products are used to find the solution, but seeing as we never used them in class, and all the textbook gives is their definition, I have no idea as to how they may be useful.

## Homework Statement

Consider the Inner Product Space P2 with the evaluation inner product at points x

_{0}=-1, x_{1}=0, and x_{2}=2, and consider the subspace W=span(**p**,_{1}**p**) where_{2}**p**=1+x+x_{1}^{2}and**p**=2-3x_{2}^{2}. Express the polynomial**p**=2+x+x^{2}in the form**w**=**w**+_{1}**w**where_{2}**w**is an element of W and_{1}**w**is an element of W orthogonal._{2}## Homework Equations

Evaluation inner product points are defined such that <

**p,q**>=p(x_{0})q(x_{0})+p(x_{1})q(x_{1})+...p(x_{n})q(x_{n})## The Attempt at a Solution

First I have to find an orthogonal basis for W. But I can't use three basis vectors because the question demands a combination of two vectors. — I have no idea where to go. I used Gram-Smidt to come up with two orthogonal bases

**v1**and**v2**. . . but I can't use just these two vectors to solve c_{1}**v**+c_{1}_{2}**v**=w._{2}And now I'm stuck. I assume evaluation inner products are used to find the solution, but seeing as we never used them in class, and all the textbook gives is their definition, I have no idea as to how they may be useful.

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