Chapter 1

Question 1

A linear equation in the variables x₁…….x_n
is an equation that can be written in the form:

Answer 1

a₁x₁ + a₂x₂ + … + a_nx_n = b

where b and the coefficients a₁…..a_n are real or complex numbers, usually known in advance. The subscript n may be any positive integer.

Question 2

A system of linear equations (or a linear system):

Answer 2

Is a collection of one or more linear equations involving the same variables—say, x₁…..x_n

Question 3

A solution of the system of linear equations:

Answer 3

Is a list (s₁, s₂…..s_n) of numbers that makes each equation a true statement when the values s₁…..s_n are substituted for x₁…..x_n respectively.

Question 4

The set of all possible solutions is called the ______ of the linear system.

Answer 4

Solution set

Question 5

Two linear systems are called ______ if they have the same solution set.

Answer 5

Equivalent

Question 6

Two linear systems are called equivalent if:

Answer 6

They have the same solution set.

Question 7

A system of linear equations is said to be ______ if it has either one solution or infinitely many solutions

Answer 7

Consistent

Question 8

When is a system of linear equations consistent?

Answer 8

If it has either one solution or infinitely many solutions

Question 9

A system is ______ if it has no solution.

Answer 9

Inconsistent

Question 10

System is inconsistent if:

Answer 10

It has no solution

Question 11

A rectangular array is a:

Answer 11

Matrix

Question 12

A ______ matrix consists only of the coefficients of the variables in a system of linear equations.

Answer 12

Coefficient

Question 13

A ______ matrix includes both the coefficients of the variables and the constants from the right-hand side of the equations.

Answer 13

Augmented

Question 14

Two matrices are called ______ if there is a sequence of elementary row operations that transforms one matrix into the other.

Answer 14

Row equivalent

Question 15

Two matrices are row equivalent if:

Answer 15

There is a sequence of elementary row operations that transforms one matrix into the other.

Question 16

True or False:

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Answer 16

True

Question 17

True or False:

If the augmented matrices of two linear systems are row equivalent, then the two systems do not have the same solution set.

Answer 17

False

Question 18

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

Answer 18

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.

Question 19

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

Answer 19

1. The leading entry in each nonzero row is 1.
2. Each leading 1 is the only nonzero entry in its column.