In addition to the topics covered on the midterm exams, the final exam will cover the following topics.
Eigenvalues and Eigenvectors Know how to find the eigenvalues of a linear transformation, as well as bases for its eigenspaces.
Diagonalizability. Know what it means for a matrix to be diagonalizable, and how to determine whether or not a given matrix is diagnonalizable.
Orthogonality Know what it means for a set of vectors to be orthogonal or orthonormal. Know the definition of an orthogonal matrix and its properties.
Orthogonal Complements and Projections Know the definition and basic properties of the orthogonal complement of a subspace of Rn. Know the definition and properties of the orthogonal projection onto a subspace of Rn.
Gram-Schmidt Process Know how to use the Gram-Schmidt process to produce an orthogonal or orthonormal basis for a given subspace of Rn.
Important Definitions/Theorems/Axioms
I will expect you to know and be able to use all of the definitions, examples and theorems below to prove results similar to those on the homework assignments. You should know precise statements of the definitions and theorems in bold.
Definition 4.1.2
Proposition 4.1.5
Definition 4.1.6
Proposition 4.1.7
Proposition 4.1.9
Definition 4.1.11
Proposition 4.1.12
Definition 4.2.1
Proposition 4.2.2
Proposition 4.2.4 and Corollar 4.2.5
Theorem 4.2.7 and Corollary 4.2.8 and Corollary 4.2.9
