Math 317 §B: Objectives for Third Hour Exam
Concepts: State the definitions and use them in proofs.
- Basis for a vector space.
- Dimension of a vector space.
- Ordered basis for a vector space.
- Coordinates of a vector with respect to a basis.
- Transition matrix for change of coordinates.
- Function, domain, codomain, range, image, pre-image; one-to-one function; onto function. Composition of functions; inverse function.
- Linear transformation; linear operator.
- Similarity of matrices.
- Kernel and range of a linear transformation.
- Isomorphism, isomorphic vector spaces.
- Eigenvalue, eigenvector of a linear operator.
- Eigenspace of an eigenvalue.
- Characteristic polynomial of a linear operator.
- Algebraic and geometric multiplicity of an eigenvalue.
- Diagonalizable linear operator.
State the Theorems and use them in Proofs.
- Use the characterizations of a basis as
- a minimal spanning set;
- a maximal linearly independent set.
- Use the fact that the image or pre-image of a subspace under a linear transformation is a subspace.
- Use the Dimension Theorem.
Apply the Algorithms and state the Theorems that justify them.
- Use row reduction to find a basis for the span of a set of vectors:
- by the Simplified Basis Method;
- by the Independence Test, to shrink a spanning set.
- Use row reduction to enlarge a linearly independent set to a basis.
- Given an ordered basis for a subspace, determine whether a vector belongs to the subspace; if it does, find its coordinates with respect to the basis.
- Find and use the transition matrix from one basis to another.
- Find the matrix of a linear transformation with respect to given bases in the domain and co-domain spaces.
- Find bases for the kernel and range of a matrix linear transformation.
- Use the matrix of a linear transformation to determine whether the transfomation is
- one to one;
- onto;
- an isomorphism.
- Compute the characteristic polynomial of a linear operator; find its eigenvalues and their eigenspaces.
- Determine whether a linear operator is diagonalizable; if possible, find a basis in which the operator is represented by a diagonal matrix.