Solution Manual For Finite Mathematics and Its
Applications, 13 Edition by Larry J. Goldstein
Linear Equation - ANSWER: in the variables x1,...xn is an equation that can be written
in the form:
a1x1 + a2x2 + anxn = b
where b and the coefficients a1, ... an are real numbers
A System of Linear Equations (Linear System) - ANSWER: a collection of one or more
linear equations involving the same variable
Is
2x1-x2+5x3 = 9
x1 - x3 = -7
a linear system? - ANSWER: Yes
Is
4x1 - 5x2 = x1x2
a linear equation? - ANSWER: No
Is
3x1 - 5x2 = -2
a linear equation? - ANSWER: yes
Two linear systems are called _____________ if they have the same solution set. -
ANSWER: equivalent
Consistent linear equation - ANSWER: - one solution
-infinitely many
Inconsistent linear equation - ANSWER: no solution
m x n matrix - ANSWER: m rows
n columns
Elementary Row Operations - ANSWER: 1. Replacement
2. Interchange
3. Scaling
Row Equivalent - ANSWER: Two matrices are row equivalent if there is a sequence of
elementary row operations that transforms one matrix into the other.
, Two Fundamental Questions About a Linear System - ANSWER: 1. Is the system
consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?
Is the system consistent? - ANSWER: Does at least one solution exist?
If a solution exists, is it the only one? - ANSWER: Is the solution unique?
Leading Entry - ANSWER: refers to the leftmost nonzero entry (in a nonzero row)
Echelon Form - ANSWER: 1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to the right of the leading entry of the
row above it
3. All entries in a column below a leading entry are zeros
Reduced Echelon Form - ANSWER: Follows all the rules of echelon form and,
1. The leading entry in each nonzero row is 1
2. Each leading 1 is the only nonzero entry in its column
Uniqueness of the Reduced Echelon Form Theorem - ANSWER: Each matrix is row
equivalent to one and only one reduced echelon matrix
Pivot Position - ANSWER: a pivot position in a matrix A is a location in A that
corresponds to the leading 1 in the reduced echelon form of A
Pivot Column - ANSWER: a pivot column is a column of A that contains a pivot
position
Basic Variable - ANSWER: a variable in a linear system that corresponds to a pivot
column in the coefficient matrix
Free Variable - ANSWER: any variable in a linear system that is not a basic variable
Existence and Uniqueness Theorem - ANSWER: a linear system is consistent if and
only if the rightmost column of the augmented matrix is NOT a pivot column - that is,
if and only if an echelon form of the augmented matrix has NO row of the form
[0 ... 0 b] with b nonzero.
If a linear system is consistent, then the solution set contains either
(i) a unique solution, when there are no free variables,
(ii) infinitely many solutions, when there is at least one free variable.
Column Vector - ANSWER: A matrix with only one column
u = [3
-1]
Applications, 13 Edition by Larry J. Goldstein
Linear Equation - ANSWER: in the variables x1,...xn is an equation that can be written
in the form:
a1x1 + a2x2 + anxn = b
where b and the coefficients a1, ... an are real numbers
A System of Linear Equations (Linear System) - ANSWER: a collection of one or more
linear equations involving the same variable
Is
2x1-x2+5x3 = 9
x1 - x3 = -7
a linear system? - ANSWER: Yes
Is
4x1 - 5x2 = x1x2
a linear equation? - ANSWER: No
Is
3x1 - 5x2 = -2
a linear equation? - ANSWER: yes
Two linear systems are called _____________ if they have the same solution set. -
ANSWER: equivalent
Consistent linear equation - ANSWER: - one solution
-infinitely many
Inconsistent linear equation - ANSWER: no solution
m x n matrix - ANSWER: m rows
n columns
Elementary Row Operations - ANSWER: 1. Replacement
2. Interchange
3. Scaling
Row Equivalent - ANSWER: Two matrices are row equivalent if there is a sequence of
elementary row operations that transforms one matrix into the other.
, Two Fundamental Questions About a Linear System - ANSWER: 1. Is the system
consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?
Is the system consistent? - ANSWER: Does at least one solution exist?
If a solution exists, is it the only one? - ANSWER: Is the solution unique?
Leading Entry - ANSWER: refers to the leftmost nonzero entry (in a nonzero row)
Echelon Form - ANSWER: 1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to the right of the leading entry of the
row above it
3. All entries in a column below a leading entry are zeros
Reduced Echelon Form - ANSWER: Follows all the rules of echelon form and,
1. The leading entry in each nonzero row is 1
2. Each leading 1 is the only nonzero entry in its column
Uniqueness of the Reduced Echelon Form Theorem - ANSWER: Each matrix is row
equivalent to one and only one reduced echelon matrix
Pivot Position - ANSWER: a pivot position in a matrix A is a location in A that
corresponds to the leading 1 in the reduced echelon form of A
Pivot Column - ANSWER: a pivot column is a column of A that contains a pivot
position
Basic Variable - ANSWER: a variable in a linear system that corresponds to a pivot
column in the coefficient matrix
Free Variable - ANSWER: any variable in a linear system that is not a basic variable
Existence and Uniqueness Theorem - ANSWER: a linear system is consistent if and
only if the rightmost column of the augmented matrix is NOT a pivot column - that is,
if and only if an echelon form of the augmented matrix has NO row of the form
[0 ... 0 b] with b nonzero.
If a linear system is consistent, then the solution set contains either
(i) a unique solution, when there are no free variables,
(ii) infinitely many solutions, when there is at least one free variable.
Column Vector - ANSWER: A matrix with only one column
u = [3
-1]
