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Section
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Content
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Suggested Problems
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Section 1.1
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- linear equation in n variables
- system of m linear equations in n variables
- solution of system of m linear equations in n variables
- solution set of system of m linear equations in n variables
- parameteric representation of solutions
- consistent and inconsistent systems of linear equations
- three types of solutions sets for systems of linear equations
- solution sets with cardinality one (unique solution)
- solution sets with infinite cardinality (infinitely many solutions)
- solution sets with cardinality zero (no solutions -- inconsistent)
- back substitution
- equivalent systems of m linear equations in n variables
- operations that produce equivalent systems
- interchange two equations
- multiply an equation by a non-zero number
- add a multiple of one equation to another equation
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Pages 10-12
1-4, 7, 9, 11-17, 27-30, 37-38, 47-48, 53-54, 67-68
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Section 1.2
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- matrix notation
- coefficient matrix of a system of m linear equations in n variables
- augmented matrix for a system of m linear equations in n variables
- elementary row operations
- interchange two rows
- multiply a row by a non-zero number
- add a multiple of one row to a second row and replace the second row
- row echelon form and reduced row echelon form
- Gaussian elimination with back substituion
- Gauss-Jordan elimination
- homoegeous systems of linear equations
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Pages 22-24
7-10, 11-14, 19-24, 25-27, 31-32, 35-36, 41-43
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Section 1.3
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- polynomial curve fitting
- translating large x-values
- network analysis
- electric circuit analysis
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Pages 32-34
3, 5, 7, 18, 27-28, 31, 33
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Section 2.1
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- definition of equality for two matrices
- definition of sum for two matrices
- definition of scalar multiplication for a scalar and a matrix
- definition of matrix product for a row vector and a column vector
- definition of product for two matrics
- matrix equation representation of a system of m linear equations in n variables
- solve a system of linear equations via matrix representation
- definition of a linear combination of column vectors
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Pages 48-51
3-4, 7-8, 11, 15, 17, 21-22, 25-26, 31-38, 41-42, 45-46, 49-50
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Section 2.2
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- algebraic properties of matrix addition and scalar multiplication
- algebraic properties of matrix multiplication
- differences between algebraic properties of real multiplication and matrix multipication
- non-commutativity of multiplication
- non-cancellation of multiplication in equations
- three types of solutions sets for systems of linear equations
- solution sets with cardinality one (unique solution)
- solution sets with infinite cardinality (infinitely many solutions)
- solution sets with cardinality zero (no solutions -- inconsistent)
- transpose of a matrix
- algebraic properties of transposes
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Pages 59-61
1, 3, 5, 7, 9, 11, 23, 25, 33-34, 37, 39, 43
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Section 2.3
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- definition of the inverse of a square matrix
- uniqueness of the inverse of a matrix
- finding the inverse of a matrix by Gauss-Jordan elimination
- algebraic properties of the inverse of a matrix
- solving a system of equations using the inverse of a matrix
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Pages 71-73
3, 5, 7, 9, 12-13, 15, 23-24, 31-33, 41, 45, 47
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Section 2.4
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- definiton of an elementary matrix
- multiplication by elementary matrices represents elementary row operations
- type I: interchanges two rows upon multiplication (on the left)
- type II: multiplies a row by a non-zero constant upon multiplication (on the left)
- type III: adds a multiple of one row to a second row and replaces the second row with the sum upon multiplication (on the left)
- multiplication of elementary matrix on the right induces action on the columns
- definition of row equivalance of matrices
- elementary matrices are invertible
- Theorem 2.15 on qquivalent conditions for Non-singularity (pg 78)
- definition of LU factorization of a matric
- construction of LU factorization via (elementary) row reduction
- solving systems of linear equations via LU factorizations
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Pages 82-83
1-8, 9-12, 13-14, 17-20, 27, 31, 35-36, 41, 43, 45
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