MATH 462, Review for Midterm Test 1

Test coverage

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 transformations and matrix multiplication (2.3)
   4. Invertible transformations, isomorphisms (2.4)
   5. Change of coordinates in linear transformations (2.5)

Important definitions

• Vector space, Subspace
• Direct sum of subpaces
• 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
• Linear isomorphism
• Similar matrices

Theorems to know with proofs

• Axioms of the vector space and their corollaries (Theorems 1.1,2 + Corollaries)
• Coordinates of a vector in a basis (Theorem 1.8)
• Every generating set can be reduced to a basis (Theorem 1.9)
• Replacement Theorem and its Corollaries (Theorem 1.10+)
• The dimension of subspaces (Theorem 1.11)
• Range of a linear transformation (Theorem 2.2)
• The Rank-Nullity theorem (Theorem 2.3)
• Theorems about one-to-one and onto mappings (2.4, 2.5)
• Property of the matrix representation (Theorem 2.14)
• The inverse of a linear transformation is linear (Theorem 2.17)
• Criterion for isomorphism (Theorem 2.19)

Important techniques

• Formal logic and proof structure as used throughout the course. In problems asking for proofs, you need to clearly state the assumptions, the conclusions, and write down a sequence of logical implications (either in words or using formal logical notation) which leads from the former to the latter.
• Reformulating problems involving vectors from other spaces than Fⁿ (polynomials, matrices, etc) in terms of matrix algebra (see for example problems 2 (1.5), 3 (1.6), 5 (2.1), etc.)
• Formulating systems of linear equations in terms of (augmented) matrices; row reduction -- check Math 262 textbook for details
• Finding bases for nullspaces of linear transformations (solving linear systems in the case of infinitely many solutions) -- follows by row reduction -- check Math 262 textbook.
• Removing redundant vectors to obtain a basis for the range; check problem 15 in 1.5, and Math 262 textbook as well.