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Mindscrape
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Let V = P^{(4)} denote the space of quartic polynomials, with the L^2 inner product
<p,q>= \int_{-l}^l p(x)q(x)dx
Let W = P^2 be the subspace of quadratic polynomials.
a) Write down the conditions that a polynomial p \in P^{(4)} must satisfy in order to belong to the orthogonal complement Wperp.
b) Find a basis for and the dimension of Wperp.
c) Find an orthogonal basis for Wperp.
The first part just goes off the definition of a complementary subspace.
W_{perp} = (q(x) = a + bx + cx^2 + dx^3 + ex^4 | <p,1>=<p,x>=<p,x^2>=0)
It looks like the second part wants me to actually do the calculation, but that looks like a lot of work to multiply it out. Is that really what it is asking?
The last part should be easy, because I can just take the basis from part b) and apply the Gram-Schmidt method.
<p,q>= \int_{-l}^l p(x)q(x)dx
Let W = P^2 be the subspace of quadratic polynomials.
a) Write down the conditions that a polynomial p \in P^{(4)} must satisfy in order to belong to the orthogonal complement Wperp.
b) Find a basis for and the dimension of Wperp.
c) Find an orthogonal basis for Wperp.
The first part just goes off the definition of a complementary subspace.
W_{perp} = (q(x) = a + bx + cx^2 + dx^3 + ex^4 | <p,1>=<p,x>=<p,x^2>=0)
It looks like the second part wants me to actually do the calculation, but that looks like a lot of work to multiply it out. Is that really what it is asking?
The last part should be easy, because I can just take the basis from part b) and apply the Gram-Schmidt method.
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