Exam 10: Inner Product Spaces

Determine , where and .

Find the Fourier approximation for .

If is an orthonormal set in an inner product space V, then is linearly independent.

If and are nonzero subspaces of an inner product space V, and v is any vector in V, then .

Let , , and let the inner product on be given by . Use the fact that is an orthogonal basis for S to find .

Find the weighted least squares line for the data set , with the inner three points weighted three times as much as the outer two points.

Determine the values of a (if any) that will make the given set of vectors orthogonal in with respect to the inner product .

Determine all values of a such that in the vectors and are orthogonal with respect to the inner product .

Use the Gram-Schmidt process to convert the set to an orthonormal basis with respect to the inner product .

Find the Fourier approximation for .

If is a linear dependent set of nonzero vectors in a vector space V, and is any inner product on V, then there exists such that .

Evaluate , where the inner product on is .

The following is an inner product on :

Use the Gram-Schmidt process to convert the set to an orthonormal basis with respect to the inner product .

Find the weighted least squares line for the data set , with the inner point weighted t times as much as the outer two points.

Find the Fourier coefficients for the function without computing any integrals.

Let be a subspace of . Let . Find with respect to the inner product .

Find the weighted least squares line for the data set , with the inner three points weighted four times as much as the outer two points.

Evaluate the given inner product on for .

Evaluate the given inner product on for .