Gram-Schmidt Process: Find Basis W for Polynomials P_2

[Hiche]

Homework Statement

The question states that we should use the Gram-Schmidt to find an orthogonal basis for W where W = span {p , q} and p(x) = 1 + x ; q(x) = 1 + 2x^2

Homework Equations

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The Attempt at a Solution

Let \{b1, b2\} be such a basis. Using the G-S process:
b1 = p = 1 + x
b2 = q - proj^q _b = q - \frac{<q, b1>} { \|b1\|^2} . b1

Is this the right way? The answer I got was \{b1, b2\} = \{1 + x, 1/2 - x/2 + 2x^2\}

Note that p and q belong to the set of polynomials of degrees 2 or less (i.e \in P_2) and < , > denotes an inner product of two components.

Also, presume I need to find further basis -- b3 -- is there a generality for bn for example?

 

[Mark44]

Homework Statement

The question states that we should use the Gram-Schmidt to find an orthogonal basis for W where W = span {p , q} and p(x) = 1 + x ; q(x) = 1 + 2x^2

Homework Equations

...

The Attempt at a Solution

Let \{b1, b2\} be such a basis. Using the G-S process:
b1 = p = 1 + x
b2 = q - proj^q _b = q - \frac{<q, b1>} { \|b1\|^2} . b1

Is this the right way? The answer I got was \{b1, b2\} = \{1 + x, 1/2 - x/2 + 2x^2\}

I haven't checked your work closely, but this is the right idea. You can check orthogonality by taking the inner product of these two functions. BTW, you didn't say what the inner product was for your space - an integral of some kind?

Note that p and q belong to the set of polynomials of degrees 2 or less (i.e \in P_2) and < , > denotes an inner product of two components.

Again, what is the inner product you are using?

Also, presume I need to find further basis -- b3 -- is there a generality for bn for example?

{b₁, b₂} is the basis for W, which is a proper subspace of P₂.

The dimension of W is 2, so any basis for W will consist of two lin. independent vectors/functions. This means it is not possible to find another basis function, which is what I think you are asking. If you add a vector or function to a basis, the new addition will necessarily be a linear combination of the other elements in the set.

The dimension of P₂ is 3, so a basis for this function space will consist of three lin. independent functions.

If you are asking whether the Gram-Schmidt process can be extended to find more than two orthogonal basis elements - yes, it can. I'm sure you can find an article on wikipedia about this.

 

[Hiche]

Oh right. The inner product is defined on P_2 as such: <p, q> = a_0b_0 + a_1b_1 + a_2b_2 where a_ns are the constants of p(x) and b_ns are the constants of q(x).