Section

Content

Suggested Problems

Section 1.1

 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 nonzero number
 add a multiple of one row to a second row and replace the second row
 pivotal row, pivotal element

1, 5, 6

Section 1.2

 row echelon form
 Gaussian elimination
 types of systems
 overdetermined
 undetermined
 reduced row echelon form
 application #1: traffic flow

1, 2, 3, 5, 7, 8, 13,

Section 1.3

 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
 noncommutativity of multiplication
 noncancellation 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 (nonsingular) matrix
 Theorem 1.3.3 (pg 49)
 transpose of a matrix
 algebraic properties of transposes
 symetric matrix

1, 2, 4, 10, 13, 22, 23, 25

Section 1.4

 elementary matrices
 type I: interchanges two rows upon multiplication (on the left)
 type II: multiplies a row by a nonzero 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 Nonsingularity (pg 65)
 construction of matrix inverse via row reduction of the maxtrix augmented with the identity
 triangular matrices
 diagonal matrices
 Theorem: Triangular matrix is nonsingular if trace is nonzero

1, 3, 4, 5, 6, 10, 13, 23

Section 2.1

 determinant of 2x2 matrix
 definition of the determinant of a matrix
 Theorem 2.1.1 (pg 95)properties of determinants
 determinant of A^{T}
 determinant of triangular matrix
 determinant of A if A has a zero row (column)
 determinant of A if A has two identical rows (columns)

1, 2, 3, 56, 11

Section 2.2

 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 nonsingular 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)

2a, 3, 4, 6, 7, 9
