MATH 462, Review for Midterm Test 2

Test topics

1. Linear transformations and matrices
   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)
2. Determinants
   1. Determinants of order n: definition and basic properties (4.2)
   2. Gauss elimination and further properties of determinants (4.3)
3. Diagonalization
   1. Eigenvalues and eigenvectors (5.1)
   2. Diagonalization (5.2, Theorem 5.5)

Important definitions

• Linear transformations, linear operators
• The space L(V;W) of linear transformations
• Matrix of a linear transformation
• Matrix product; invertible transformation; inverse matrix
• Isomorphism of vector spaces
• Determinant of a matrix (cofactor expansion)
• Elementary matrices
• Gauss elimination; Reduced row echelon form
• Rank of a matrix and rank of a linear transformation
• Eigenvectors, eigenvalues, eigenspace
• Diagonalizable operator

Important theorems

• Composition and the matrix product (Theorem 2.11)
• Inverse of a linear transformation is linear (Theorem 2.17)
• Dimension criterion for isomorphism of vector spaces (Theorem 2.19)
• Isomorphism between linear transformations and matrices (Theorem 2.20)
• Change of coordinates (Theorem 2.23)
• Properties of determinants (Corollary 4.4, Theorems 4.5 and their equivalence; Theorem 4.6)
• Determinants of matrix products and transposes (Theorems 4.7, 4.8)
• Criterion for diagonalizability (Theorem 5.1)
• Linear independence of eigenvectors (Theorem 5.5)

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