Section

Content

Suggested Problems

Section 2.5

 Stochastic Matrices
 States
 Matrix of transition probabilities
 Least Squares Regression
 Definition of least squares regression line
 Matrix form for linear regression

Pages 9597
5, 7, 9, 35, 37, 39

Section 3.1

 Determinant of 2x2 matrix
 Minors and cofactors of a square matrix
 Definition of the determinant: Row one cofactor expansion
 Theorem3.1: Computation of the determinant of matrix by any row or any column cofactor expansion
 Aternate method for computation of the determinant of a 3x3 matrix
 Determinants of triangular matrices

Pages 110111
57, 1921, 2728, 3334, 3941, 47, 51

Section 3.2

 Elementary Row Operations and Determinants
 Type I: B obtained from A by interchanging two rows > det(B) = det(A)
 Type II: B obtained from A by multipling a row of A by a nonzero constant c > det(B) = c det(A)
 Type III: B obtained from A by adding a multiple of one row to a second row and replacing the second row with the sum > det(B) = det(A)
 Finding a determinant using elementary row operations
 Determinants and elementary column operations
 Theorem 3.4: Conditions that yield zero determinants
 if A has a zero row (column)
 if A has two identical rows (columns)
 Finding determinants

Pages 118119
25, 27, 29, 31, 33, 35, 3942

Section 3.3

 Determinant of a matrix product det(AB) = det(A) det(B)
 Determinant of a scalar multiple of a matrix
 Determinant of an invertible matrix
 Determinant of the inverse of a matrix
 Determinant of the transpose of a matrix

Pages 125127
3, 9, 13, 1720, 2627, 3739, 53, 55

Section 3.4

 Adjoint of a matrix
 Matrix of cofactors
 Inverse of matrix via its adjoint
 Cramer's Rule
 Area in the plane
 Area of triangle
 Test for colinearity
 Twopoint form of the equation of a line
 Volume in space
 Volume of a tetrahedron
 Test for coplanarity
 Threepoint form of the equation of a plane

Pages 136137
1, 3, 17, 19, 25, 27, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59

Section 4.1

 Vectors in the plane and space
 Vector addition
 Geometric definition
 Algebraic definition
 Scalar multiplication
 Geometric definition
 Algebraic definition
 Vector addition and scalar multiplication properties
 binary operation addition
 closed
 commutative
 associative
 existence of an additive identity (called 0)
 existence of additive inverse for each a (called a)
 scalar multiplication
 closed
 distributive property i
 distributive property ii
 associativity of scalar multiplication
 unity property of scalar multiplication
 Vectors in R^{n}
 Vector addition and scalar multiplication
 Vector addition and scalar multiplication properties
 Properties of additive identity and additive inverse
 Linear combination of vectors

Pages 153154
710, 1924, 3738, 3940, 4546, 5354

Section 4.2

 Definition of a vector space
 binary operation addition
 closed
 commutative
 associative
 existence of an additive identity (called 0)
 existence of additive inverse for each a (called a)
 scalar multiplication
 closed
 distributive property i
 distributive property ii
 associativity of scalar multiplication
 unity property of scalar multiplication
 Standard examples of vector spaces
 Properties of scalar multiplication
 Testing sets with operations to determine whether they form vector spaces

Pages 160161
13, 15, 1718, 21, 23, 2526, 2930, 35

Section 4.3

 Definition of a subspace
 Testing subsets of vector spaces to determine whether they form a subspace
 Theorem 4.6: The intersection of two subspaces is a subspace
 Subspaces of R^{2} and of R^{3}

Pages 167168
712, 21, 23, 25, 29, 31, 33, 37, 39, 41

Section 4.4

 Definition of linear combination
 Definition of a spannng set
 Testing sets to determine whether they span a vector space
 Defintion of the span of a set
 Theorem 4.7: The span of a set is a subspace
 Definition of linear independence and linear dependence
 Testing sets to determine whether they are linearly independent
 Sets containing two vectors
 Subsets of R^{n}
 Subsets of P_{n}

Pages 178179
12, 56, 11, 13, 15, 21, 23, 25, 29, 32, 35, 38, 4143, 45, 46

Section 4.5

 Definition of basis B
 B is a spanning set
 B is linearly independent
 R^{n}: Standard basis & nonstandard basis
 P_{n}: Standard basis & nonstandard basis
 M_{m,n}: Standard basis & nonstandard basis
 Theorem 4.9: Uniqueness basis representation
 Theorem 4.10 Basis and linear dependence
 Theorem 4.11 Every basis of a vector space V has the same number of elements
 Definition of dimension of a vector space
 Theorem 4.12 Basis tests in ndimensional vector space

Pages 187188
710, 1518, 2528, 33, 35, 37, 41, 43, 45

Section 4.6

 Row space of a matrix A
 Column space of matrix A
 Theorem 4.13: Rowequivalent matrices have the same row space
 Theorem 4.14: Mechanism for finding a basis for the row space of a matrix
 Find a basis for the row space of a matrix A
 Reduce A to RREF
 The rows with leading ones of the RREF form a basis for the row space of A
 Find a basis for a subspace
 Cold Hard Fact: The column space of a matrix is destroyed under rowequivalent operations
 Find a basis for the column space of a matrix A
 Apply elementary column operations to A
 Take the transpose of A and apply elementary row operations to A^{T}
 Elementary row operations on A preserve the dependency relationships between the columns of A
 Reduce A to RREF
 The columns of A which correspond to the columns of RREF which contains leading ones form a basis for the column space of A
 Theorem 4.15: dim(row space of A) = dim(col space of A)
 Definition of the rank of a matrix A
 Theorem 4.16: The set of solutions of Ax = 0 forms a subspace of R^{n}
 Definition of the null space of a matrix A
 Definition of the nullity of a matrix A
 Finding the nullspace of a matrix A
 Finding a basis for the nullspace of a matrix A
 Dimension theorem: Theorem 4.17
 Solutions of linear systems of equations
 Solutions of the homogeneous equation Ax = 0
 Solutions of the nonhomogeneous equation Ax = b
 Theorem 4.19: Solution set of Ax = b is same as the column space of A
 Equivalency of conditions for a matrix A to be invertible

Pages 199201
7, 9, 11, 13, 15, 17, 21, 23, 29, 31, 33, 39, 41, 43, 5758
