Exam 10: Inner Product Spaces

In a weighted least squares regression, the average of the weights must equal 1.

If V is a finite-dimensional inner product space, S is a nonzero subspace of V and is nonzero, then for every in V, .

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

If f is a positive even function on , then for every k, and in the Fourier approximation of f.

Find the Fourier approximation for the odd function .

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

Evaluate the given inner product on for . .

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

Find , where , where and the inner product on is .

If is an inner product on a vector space V, and v is any vector in V, then defined by is a linear transformation.

Determine , where and .

If is an inner product on a vector space V, and is an invertible linear transformation, then is also an inner product on V.

Find the discrete Fourier approximation for based on the table information.

If f is an odd function on , then in the Fourier approximation of f, we have for every k.

If is an orthonormal set in an inner product space V, and is a orthogonal matrix, then is also an orthonormal set in V.

Evaluate , where the inner product on is for .

If is the nth-order Fourier approximation to f, and is the nth-order Fourier approximation to g, and are scalars, then the nth-order Fourier approximation to is .

If u, v are vectors in an inner product space V, then u is orthogonal to v if and only if .

Evaluate the given inner product on for .

Find the discrete Fourier approximation for based on the table information.