MATH 462, Review for the Final Exam

Course topics

1. Vector Spaces
   1. Axioms of the vector space (1.2)
   2. Subspaces (1.3)
   3. Linear combinations, systems of linear equations (1.4)
   4. Linear dependence and linear independence (1.5)
   5. Bases and dimension (1.6)
2. Linear transformations and matrices
   1. Linear transformatitions. Null spaces and ranges (2.1)
   2. The matrix representation of a linear transformation (2.2)
   3. Composition of linear transformations and matrix multiplication (2.3)
   4. Invertibility and isomorphisms (2.4)
   5. Change of coordinates for vectors and linear transformations (2.5)
3. Determinants
   1. Determinants of order n: definition and basic properties (4.2)
   2. Gauss elimination and further properties of determinants (4.3)
4. Diagonalization
   1. Eigenvalues and eigenvectors (5.1)
   2. Diagonalization (5.2)
   3. Invariant subspaces and the Caley-Hamilton theorem (5.4)
5. The Jordan canonical form
   1. The Jordan canonical form (7.1)
   2. Uniqueness and examples (7.2)

Important definitions (in addition to Midterms 1 and 2)

• Invariant subspace
• Cyclic subspace
• The Jordan canonical form
• The Jordan basis
• Generalized eigenvector, generalized eigenspaces
• Cycle of generalized eigenvectors

Important theorems (cumulative list)

• Axioms of the vector space and their corollaries
• Replacement theorem (Theorem 1.10) and its corollaries
• The rank-nullity theorem (Theorem 2.3)
• Vector spaces are isomorphic iff they have the same dimension (2.19 and Lemma preceding Theorem 2.18)
• Isomorphism between linear transformations and matrices (Theorem 2.20)
• Criterion for diagonalizability (Theorem 5.1)
• Linear independence of eigenvectors (Theorem 5.5)
• The algebraic and geometric multiplicity (Theorem 5.7)
• Invariant subspaces and characteristic polynomials (Theorem 5.21)
• Basis of a cyclic subspace (Theorem 5.22)
• The Caley-Hamilton theorem (Theorem 5.23)
• Properties of generalized eigenvectors (Theorems 7.1,2)

A number of questions on the exam will be simple yes/no type questions, which may be taken from Problems 1 in each section.