Section

Content

Suggested Problems

Section 3.1

 vector space (linear space)
 binary operation addition
 commutative
 associative
 existence of an additive identity (called 0)
 existence of additive inverse for each a (called a)
 scalar multiplication
 distributive property i
 distributive property ii
 associativity of scalar multiplication
 unity property of scalar multiplication
 Example: R^{n}
 addition = componentwise addition
 scalar multiplication = componentwise scalar multiplication
 R^{2}, R^{3}
 vectors identified with "directed line segments"
 componentwise addition identified with geometry of parallolegram rule
 scalar multiplication identified with geometric scaling of vector
 length of vector given by Pathagorean rule
 Example: R^{nxm}
 addition = matrix addition
 scalar multiplication = scalar multiplication on matrices
Example: P_{n}
 addition = polynomial addition
 scalar multiplicaiton = multiplication of scalars on polynomials
 Example: C[a,b}
 addition = function addition
 scalar multiplicaiton = multiplication of scalars on functions
 Theorem 3.1.1 (Additional Properties of Vector Spaces)

4, 6, 8, 9, 10, 11, 12

Section 3.2

 definition of a subspace
 examples for R^{2}
 examples for R^{2x2}
 examples for P_{3}
 null space of a matrix N(A)
 span of a set of vectors
 Theorem 3.2.1
 spanning set for a vector space V

1, 2, 3, 5, 7, 9, 10, 11, 14, 17

Section 3.3

 linear independence
 linear dependence
 linear independence of two vectors
 Theorem 3.3.1 Linear independence of n vectors in R^{n}
 Theorem 3.3.2
 Linear Independence Conditions/Criteria
 R^{n}
 P_{n}
 R^{nxm}
 C^{n1}[a,b]

1, 2, 6, 7, 11

Section 3.4

 basis
 Examples
 R^{2}
 R^{3}
 R^{2x2}
 P_{3}
 Theorem 3.4.1
 Corolary 3.4.2
 dimension
 finite dimensional vector space
 infinite dimensional vector space
 Theorem 3.4.3
 Standard Bases

1, 2, 3, 5, 7, 10, 11, 14

Section 3.5

 R^{n}
 Standard Basis for R^{n}
 Ordered Basis in R^{n}
Coordinates w.r.t Ordered Basis
 Problem
 1. Given coordinates in standard basis find coordinates in ordered basis
 2. Given coordinates in ordered basis find coordinates in standard basis
 Solution
 Problem 2: Transition Matrix from Ordered Basis to Standard Basis U
 Problem 1: Transition Matrix from Standard Basis to Ordered Basis U^{1}
 Transition Matrix S from Ordered Basis F to Ordered Basis E
 General Finite Dimensional Vector Space
 Ordered Basis
Coordinates w.r.t Ordered Basis
 Problem
 Given coordinates in ordered basis E find coordinates in ordered basis F
 Solution
 Problem 2: Transition Matrix S from Ordered Basis F to Ordered Basis E

1, 2, 3, 5, 6, 10

Section 3.6

 vector space (linear space)
 Row Space of A
 Column Space of A
 Theorem: Two row equivalent matrices have the same row space
 Theorem 3.6.2 Consistency Theorem for Linear Systems
 Theorem: A (nxn) is nonsingular iff the column space of A = R^{n}
 Rank of A
 Nullity of A
 Theorem 3.6.5. RankNullity Theorem
 MetaTheorem: Two row equivalent matrices have different column spaces, but the same column dependency relationships
 Theorem 3.6.6 dim(row space of A) = dim(col space of A)

1, 2, 4, 7, 8, 10, 11, 12
