MATH 462, Review for Midterm Test 2

Test topics

1. Determinants
   1. Determinants of order n: definition and basic properties (4.2)
   2. Gauss elimination and further properties of determinants (4.3)
2. Diagonalization
   1. Eigenvalues and eigenvectors (5.1)
   2. Diagonalizability (5.2)
   3. Invariant subspaces and the Caley-Hamilton theorem (5.4)
   4. Direct sums of subspaces. Direct sums of invariant subspaces and matrices (5.2, 5.4)

Important definitions

• Ordered basis, union of ordered bases
• Linear transformations, linear operators
• Matrix of a linear transformation
• Change of basis, similarity of linear operators and matrices
• Determinant of a matrix (cofactor expansion)
• Elementary matrices
• Gauss elimination; Reduced row echelon form
• Row and column rank of a matrix; rank of a linear transformation
• Eigenvectors, eigenvalues, eigenspaces
• Diagonalizable operators
• Invariant subspace, cyclic subspace, restriction of an operator onto an invariant subspace
• Direct sums for more than two subspaces

Important theorems

• Properties of determinants (Theorems 4.4, 4.5, 4.7, 4.8)
• Linear independence of eigenvectors (Theorem 5.5)
• Algebraic and geometric multiplicity (Theorem 5.7)
• Criterion of diagonalizability (Theorems 5.9, 5.8, Lemma on p. 267)
• Basis for a cyclic subspace (Theorem 5.22)
• The Caley-Hamilton theorem (Theorem 5.23)
• Equivalent definitions of direct sums (Theorem 5.10)

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