MATH 462, Review for Midterm Test 1

Test 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)

Important definitions

• Vector space, Subspace, Direct sum, Quotient space
• Linear combination, Linear span
• Linear relation, Linearly dependent set, Linearly independent set
• Generating set, Basis, Coordinates
• Linear transformation, Range, Null space, Rank, Nullity
• Projection onto a subspace along another subspace
• Matrix representation of a linear transformation
• Isomorphism, Isomorphic spaces
• Linear functional, Dual space, Dual basis, Transpose of a linear transformation, Double dual

Important theorems

• Axioms of the vector space and their corollaries
• Coordinates of a vector in a basis (Theorem 1.8)
• Replacement theorem (Theorem 1.10) and its corollaries
• The rank-nullity theorem (Theorem 2.3)
• Theorems about one-to-one and onto mappings (2.4, 2.5)
• A linear transformation is uniquely defined by its values on the vectors of a basis (Theorem 2.6)
• Matrix representations (Theorems 2.14 and 2.11)
• Theorems about isomorphisms (2.17, 2.19, 2.20)
• The dual basis (Theorem 2.24)
• The natural isomorphism of V and V** (Theorem 2.26)

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