Exam 8: Orthogonality

Suppose S is a subspace of , and . Determine .

Find the normal equations for the given system.

Find all values of a so that u and v are orthogonal.

If a matrix A has linearly independent columns, then for every vector y there exists a unique least squares solution of .

Find the vector in the subspace S that is closest to y.

Express the given matrix as an outer product expansion , where and are the singular values of .

Determine , where .

The eigenvalues for the symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Find an orthogonal basis for the subspace .

If is an orthonormal set, then is linearly independent.

If is a linearly independent set of vectors, then is orthogonal.

Find a singular value decomposition for the matrix .

Find the singular values for the matrix .

Determine all values of a so that the vector u is orthogonal to the subspace S.

Find all values of a so that u and v are orthogonal.