Section

Content

Suggested Problems

Section 2.3

 adjoint of an nxn matrix
 Cramer's Rule

1, 2, 6

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
