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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
- three types of solutions sets
- solution sets with cardinality one (unique solution)
- solution sets with infinite cardinality (infinitely many solutions)
- solution sets with cardinality zero (no solutions -- inconsistent)
- equivalent systems of m linear equations in n variables
- strict triangular system of m linear equations in n variables
- back substitution
- coefficient matrix of 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
- pivotal row, pivotal element
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1, 5, 6
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Section 1.2
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- row echelon form
- Gaussian elimination
- types of systems
- overdetermined
- undetermined
- reduced row echelon form
- application #1: traffic flow
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1, 2, 3, 5, 7, 8, 13,
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Section 1.3
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- matrix notation
- vectors
- definition of equality for two matrices
- definition of sum for two matrices
- definition of scalar multiplication for a scalar and a matrix
- algebraic properties of matrix addition
- definition of matrix product for a row vector and a column vector
- definition of product for two matrics
- 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
- powers of matrices (nxn)
- matrix equation representation of a system of m linear equations in n variables
- definition of a linear combination
- Consistency Theorem for Systems of Linear Equations (pg 37)
- invertible (non-singular) matrix
- Theorem 1.3.3 (pg 49)
- transpose of a matrix
- algebraic properties of transposes
- symetric matrix
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1, 2, 4, 10, 13, 22, 23, 25
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Section 1.4
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- elementary matrices
- 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
- row equivalance of matrices
- Theorem Equivalent conditions for Non-singularity (pg 65)
- construction of matrix inverse via row reduction of the maxtrix augmented with the identity
- triangular matrices
- diagonal matrices
- Theorem: Triangular matrix is non-singular if trace is non-zero
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1, 3, 4, 5, 6, 10, 13, 23
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Section 2.1
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- determinant of 2x2 matrix
- definition of the determinant of a matrix
- Theorem 2.1.1 (pg 95)properties of determinants
- determinant of AT
- determinant of triangular matrix
- determinant of A if A has a zero row (column)
- determinant of A if A has two identical rows (columns)
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1, 2, 3, 5-6, 11
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Section 2.2
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- Theorem 2.2.1 (pg 98)
- determinants of elementary matrices E and determinants of multiplication (on the left) by E
- type I: det(E) = -1; det(EA) = -det(A)
- type II: det(E) = alpha; det(EA) = alpha det(A)
- type III: det(E) = 1; det(EA) = det(A)
- Theorem 2.2.2 A is non-singular iff det(A) not equal 0
- computation (simplificiations) of determinants via reductions (eliminations) by elementary matrix operations
- Theorem 2.2.3 det(AB) = det(A) det(B)
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2a, 3, 4, 6, 7, 9
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Section 2.3
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- adjoint of an nxn matrix
- Cramer's Rule
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1, 2, 6
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