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Please show all your work! Total points: (longer than an actual exam)

- Consider the space P
_{2}with inner product defined as <p(x),q(x)> = p(0)q(0). Is this an inner product space?*No, does not satisfy positivity: p(x) = x has zero length according to this inner product, but p(x) ≠ 0.* - Consider the space F[0,∞) with inner product defined as
<f,g> = ∫
_{0}^{∞}f(x)g(x)dx.- Prove that this is an inner product space.
*It is okay to assume that F[0,1] is a vector space, and just prove the four axioms on the inner product.* - Let f
_{n}(x) = e^{-nx}. Find the norm of f_{n}for any n>0.*∫ e*^{-2nx}dx = (-1/2n) e^{-nx}.

Eval at [0,∞): ||f_{n}||^{2}= 1/2n

So ||f_{n}|| = 1/√(2n). - What about f
_{n}for n≤0? Describe the graph of the functions, and find their norms.*At n=0, f*_{0}= 1 (constant function), which has infinite norm in this space. For n<0, f_{n}increases without bound (exponential growth), so the norm is also infinite. - Apply the Gram-Schmidt process to obtain an orthonormal
set from the set {f
_{1}, f_{2}}.*Let v*_{1}= f_{1}; according to a previous part, its norm is 1/√(2n) = 1/√2, so the normalized version is q_{1}= √2 e^{-x}. Is f_{2}orthogonal to q_{1}? Their inner product is √2 ∫ e^{-3x}dx = √2 /3 ≠ 0, so no.

The projection of f_{2}onto q_{1}is (2/3)e^{-x}, so the orthogonal complement is v_{2}= f_{2}- proj(f_{2}onto q_{1}) = e^{-2x}- (2/3)e^{-x}.

To normalize this, we need to find its norm:

||v_{2}||^{2}= ∫ ( e^{-2x}- (2/3)e^{-x})^{2}dx = ∫ ( e^{-4x}- (4/3)e^{-3x}+ (4/9)e^{-2x}) dx = (1/4) - (4/9) + (2/9) = 1/36.

So ||v_{2}|| = 1/6, so q_{2}= 6v_{2}= 6 e^{-2x}- 4 e^{-x}.

The orthonormal set is {q_{1}, q_{2}} = { √2 e^{-x}, 6 e^{-2x}- 4 e^{-x}}. - Is this set an orthonormal basis?
*Not for all of F[0,∞), which is an infinite-dimensional space.*

- Prove that this is an inner product space.
- Find a QR-decomposition for the matrix, if possible:
1 0 1 -1 1 1 1 0 1 -1 1 1 - Consider the set of points (0,1), (1,3), (2,2), (3,3).
- Find the polynomial of lowest degree which interpolates these points.
- Find the line which best fits these points using least-squares approximation.
- What is the error in this approximation? (i.e., ||Ax-b||).
- Find the quadratic (parabola) which best fits these points using least-squares approximation.
- What is the error in this approximation? (i.e., ||Ax-b||).

- Consider the matrix A:
1 0 0 0 -1 2 0 2 2 - Does there exist an orthogonal matrix P for which P
^{-1}AP is a diagonal matrix?*Yes, A is orthogonally diagonalizable, because it is symmetric.* - Find all eigenvalues and eigenvectors of A.
*@λ=1: x = (1,0,0)t. @λ=3: x = (0,1/2,1)t. @λ=-2: x = (0,-1/2,1)t.* - Find an orthogonal matrix P which diagonalizes A.
*Just normalize the eigenvectors:*1 0 0 0 1/√5 -1/√5 0 2/√5 2/√5

- Does there exist an orthogonal matrix P for which P