Section
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Content
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Suggested Problems
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Section 2.5
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- Stochastic Matrices
- States
- Matrix of transition probabilities
- Least Squares Regression
- Definition of least squares regression line
- Matrix form for linear regression
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Pages 95-97
5, 7, 9, 35, 37, 39
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Section 3.1
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- Determinant of 2x2 matrix
- Minors and co-factors of a square matrix
- Definition of the determinant: Row one co-factor expansion
- Theorem3.1: Computation of the determinant of matrix by any row or any column co-factor expansion
- Aternate method for computation of the determinant of a 3x3 matrix
- Determinants of triangular matrices
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Pages 110-111
5-7, 19-21, 27-28, 33-34, 39-41, 47, 51
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Section 3.2
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- 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 non-zero 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
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Pages 118-119
25, 27, 29, 31, 33, 35, 39-42
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Section 3.3
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- 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
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Pages 125-127
3, 9, 13, 17-20, 26-27, 37-39, 53, 55
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Section 3.4
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- Adjoint of a matrix
- Matrix of co-factors
- Inverse of matrix via its adjoint
- Cramer's Rule
- Area in the plane
- Area of triangle
- Test for co-linearity
- Two-point form of the equation of a line
- Volume in space
- Volume of a tetrahedron
- Test for co-planarity
- Three-point form of the equation of a plane
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Pages 136-137
1, 3, 17, 19, 25, 27, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59
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Section 4.1
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- 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 Rn
- Vector addition and scalar multiplication
- Vector addition and scalar multiplication properties
- Properties of additive identity and additive inverse
- Linear combination of vectors
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Pages 153-154
7-10, 19-24, 37-38, 39-40, 45-46, 53-54
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Section 4.2
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- 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
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Pages 160-161
13, 15, 17-18, 21, 23, 25-26, 29-30, 35
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Section 4.3
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- 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 R2 and of R3
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Pages 167-168
7-12, 21, 23, 25, 29, 31, 33, 37, 39, 41
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Section 4.4
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- 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 Rn
- Subsets of Pn
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Pages 178-179
1-2, 5-6, 11, 13, 15, 21, 23, 25, 29, 32, 35, 38, 41-43, 45, 46
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Section 4.5
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- Definition of basis B
- B is a spanning set
- B is linearly independent
- Rn: Standard basis & non-standard basis
- Pn: Standard basis & non-standard basis
- Mm,n: Standard basis & non-standard 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 n-dimensional vector space
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Pages 187-188
7-10, 15-18, 25-28, 33, 35, 37, 41, 43, 45
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Section 4.6
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- Row space of a matrix A
- Column space of matrix A
- Theorem 4.13: Row-equivalent 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 row-equivalent 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 AT
- 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 Rn
- 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 non-homogeneous 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
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Pages 199-201
7, 9, 11, 13, 15, 17, 21, 23, 29, 31, 33, 39, 41, 43, 57-58
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