Math 317 §B: Objectives for Third Midterm Exam
This document incorporates the Objectives
for the First Midterm Exam and the Objectives
for the Second Midterm Exam.
State the Definitions and use them in proofs.
- Orthogonal projection of a vector onto a subspace.
- Least squares solution of an inconsistent linear system.
- Orthogonal set of vectors.
- Orthogonal basis for a subspace.
- Orthonormal basis for a subspace.
- Orthogonal matrix.
- Linear transformation.
- Kernel of a linear transformation.
- Image of a linear transformation.
- Standard matrix of a linear transformation.
- Matrix of a linear operator with respect to a basis.
- Similar matrices.
- Isomorphism.
- Matrix of a linear transformation with respect
to bases in the domain and range spaces.
- Determinant.
- Cofactor.
State the Theorems and use them in proofs.
- Orthogonal projection onto a subspace via the normal equations.
- An orthogonal set of nonzero vectors is linearly independent.
- The orthogonal projection onto a subspace is the sum of
rank-one projections onto the vectors of an orthogonal basis
for the subspace.
- Change-of-Basis Formula.
- Effect of row operations on the determinant.
- The determinant of a product is the product of the determinants.
- If A is nonsingular, then
det(A-1) = 1/det A.
- A matrix and its transpose have the same determinant.
- A matrix is nonsingular if and only if its determinant is nonzero.
- Expansion of a determinant in cofactors.
- Cramer's Rule.
- Cofactor formula for the inverse of a matrix.
- Determinants and signed area and volume.
Algorithms: Do the following.
- Compute the least squares fit to a table of data.
- Given a basis for a subspace, apply the Gram-Schmidt
algorithm to obtain an orthogonal basis for the same
subspace.
- Compute the QR decomposition of a matrix.
Use the QR decomposition to solve least squares problems.
- Use Cramer's Rule to represent a component of the solution
of a linear system as a quotient of determinants.
