Math 317 §B: Objectives for Final Exam
This document incorporates the objectives for the hour exams too:
Concepts: State and use definitions of the following.
- In Rn or Cn or any real or complex inner product space:
- Inner product, norm of a vector, distance between vectors
- Orthogonal set of vectors, orthonormal set of vectors.
- Orthogonal basis for a vector space; orthonormal basis for a vector space.
- Orthogonal complement of a subspace.
- Orthogonal projection of a vector onto a subspace.
- Orthogonal matrix, Unitary matrix
- Symmetric operator, normal operator, Hermitian operator
- Least squares solution of Ax=b, normal equations.
Algorithms and Proofs.
- Use the fact that nonzero orthogonal vectors are linearly independent.
- Find the coordinate vector (coordinatization) of a given vector with respect to an orthogonal basis.
- Use the Gram-Schmidt algorithm to find an orthogonal basis for the span of a set of linearly independent vectors.
- Use the criteria for an n by n real matrix U to be orthogonal (and the analogous criteria for an n by n complex matrix to be unitary):
- UTU = I.
- The columns of U form an orthonormal basis of Rn.
- The rows of U form an orthonormal basis of Rn.
- U preserves dot products.
- U preserves norms.
- Find the orthogonal projection of a vector onto a subspace.
- Find the unique decomposition of a vector into the sum of a vector in a subspace W and a vector in Wperp.
- Use orthogonal projection to find the point in a subspace nearest to a given point.
- Use the spectral theorem for symmetric operators.
- Given a symmetric matrix A find an orthogonal matrix U such that UTAU is diagonal.
- Apply the Least Squares Theorem.
- Use the normal equations to find the least squares solution of a system A x = b with a matrix A of full rank.
- Find the least squares polynomial of specified degree fitting a table of data.
