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)
6. Matrix limits and Markov chains
   1. Matrix limits. Probability vectors and stochastic matrices. The matrix exponential (7.2, problem 19 and 5.3)

Important definitions (in addition to Midterms 1 and 2)

• The Jordan canonical form
• The Jordan basis
• Generalized eigenvector, generalized eigenspaces
• Cycle of generalized eigenvectors
• Invariant subspace
• Cyclic subspace
• Matrix limits
• Probability vectors and stochastic matrices
• Matrix exponential

Theorems to know with proofs (complete list)

• Replacement theorem (Theorem 1.10) and its corollaries
• The rank-nullity theorem (Theorem 2.3)
• Criterion for isomorphism (2.19 and Lemma preceding Theorem 2.18)
• Criterion for diagonalizability (Theorem 5.1)
• Linear independence of eigenvectors (Theorem 5.5)
• The algebraic and geometric multiplicity (Theorem 5.7)
• Basis of a cyclic subspace (Theorem 5.22 and problems 5.4: 19 or 4.3: 24)
• The Caley-Hamilton theorem (Theorem 5.23)
• Properties of generalized eigenvectors (Theorems 7.1, 7.2)
• Theorem 7.3 in the case of two distinct eigenvalues
• Theorems 7.6 and 7.7 about the Jordan basis

Important problem types / technical skills

• Compute an n x n determinant
• Find a matrix of a linear operator using a basis in the space of polynomials, matrices, functions, etc.
• Find the complete set of solutions for a system of linear equations
• Find eigenvalues, eigenvectors of a matrix, diagonalize
• Find the Jordan canonical form of a matrix/operator
• Find invariant subspaces of a matrix/operator
• Compute a cyclic subspace for a vector
• Compute a matrix limit / matrix exponential

The final be cumulative, but more emphasis will be put on the material from Chapters 5 and 7 (topics IV to VI on the list above). One of the questions will address the material of the last section (5.3).