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. The change of coordinates matrix (2.5)
   6. Dual spaces (2.6)
3. Determinants
   1. The area function and determinant (4.1)
   2. Definition of the determinant and the basic properties (4.2)
   3. Characterization of the determinant (4.5: Theorem 4.12)
   4. Determinants of matrix products and transposes (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. Theory (7.1)
   2. Uniqueness and examples (7.2)
6. Inner product spaces
   1. Inner products and norms (6.1)
   2. Orthogonal, orthonormal sets, bases; Orthogonal complement (6.2)
   3. The adjoint of a linear operator (6.3)

Important definitions (in addition to Midterms 1 and 2)

• Inner product, norm, inner product space
• Orthogonal, orthonormal vectors/set/basis
• Orthogonal complement
• Adjoint operator

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)
• A linear transformation is uniquely defined by its values on the vectors of a basis (Theorem 2.6)
• Vector spaces are isomorphic iff they have the same dimension (2.19 and Lemma preceding Theorem 2.18)
• Characterization of determinant (Theorem 4.12 and related facts)
• 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)
• The vector space V is a direct sum of generalized eigenspaces (Theorem 7.3) (w/o proof)
• A generalized eigenspace has a basis which is a union of cycles (Theorem 7.7) (w/o proof)
• Properties of the inner product (Theorem 6.1)
• The Cauchy-Schwarz and the triangle inequalities (Theorem 6.2)
• The orthogonal projection theorem (Theorem 6.6 and Corollary)
• The finite-dimensional Riesz' theorem (Theorem 6.8)
• The definition of the adjoint (Theorem 6.9)

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