Math 317 §B: Objectives for Third Hour Exam

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.

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.

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.