Section

Content

Suggested Problems

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

Section 4.1

 Examples
 Properties
 L(0_{V}) = 0_{W}
 L(x) = L(x)
 Linear combinations
 Kernel of L
 Image of a Subspace under L
 Range of L
 Theorem 4.1.1

1, 4, 5, 6, 7, 9, 11, 17, 19

Section 4.2

 Theorem 4.2.1 Matrix representation of Linear Transformation from R^{n} to R^{m}
 Theorem 4.2.2 Matrix representation of Linear Transformation from V to W
 Theorem 4.3.3 Construction of Matrix A for case of V = R^{n} and W = R^{m} in Theorem 4.2.2
 Corollary Above construction via row equivalent transformation of an augmented matrix

1, 2, 3, 4, 6, 14, 18

Section 4.3

 Linear Operator
 Theorem 4.3.1
 Similarity

1, 3, 4, 5
