Chapter 1: Systems of linear equations

Question 1

Linear equation in the n variables x_1, x_2,…,x_n

Answer 1

An equation of the form a_1x_1 + a_2x_2 + · · · a_nx_n = b

Question 2

Coefficients of x_i

Answer 2

a_i ∈ R in a_1x_1 + a_2x_2 + · · · a_nx_n = b

Question 3

Constant term

Answer 3

b ∈ R in a_1x_1 + a_2x_2 + · · · a_nx_n = b

Question 4

Solution of linear equation

Answer 4

a sequence of n numbers s1,s2, · · · ,sn so that
a1s1 + a2s2 + · · · ansn = b

Question 5

System of linear equations

Answer 5

a finite set of linear equations

Question 6

Solution to a system of linear equations

Answer 6

a sequence of numbers that is simultaneously a solution to all equations in the system

Question 7

Consistent system of equations

Answer 7

if it has at least one solution

Question 8

Inconsistent system of equations

Answer 8

if it has no solutions

Question 9

Homogenous system of linear equations

Answer 9

if and only if the constant term in each equation is zero.

Question 10

Equivalent systems of linear equations

Answer 10

have the same set of solutions

Question 11

ERO I

Answer 11

I Interchange two rows.

Question 12

ERO II

Answer 12

II Multiply one row by a non-zero number.

Question 13

ERO III

Answer 13

III Add a non-zero multiple of one row to a different row

Question 14

A matrix is in row-echelon form if and only if

Answer 14

1 All zero rows (consisting entirely of zeros) are at the bottom.
2 The first non-zero entry from the left in each non-zero row is a 1, called a leading 1 for that row.
3 Each leading 1 is to the right of all leading 1s in the rows above it.

Question 15

A matrix is said to be in reduced row-echelon form if, in addition to the conditions for REF, it satisfies the following condition:

Answer 15

4 Each leading 1 is the only non-zero entry in its column.

Question 16

Gaussian elimination

Answer 16

The step-by-step procedure for finding a (reduced) row-echelon matrix

Question 17

The reduced row-echelon form of a matrix A is ______

Answer 17

unique

Question 18

the number of leading 1s must be ___ ____ in each row-echelon form of A.

Answer 18

the same

Question 19

Rank of a matrix A

Answer 19

the number of leading 1s in any row-echelon form of the matrix A.

Question 20

Theorem 1.2.2 Suppose a system of m equations in n variables is consistent and that the rank of the augmented matrix is r. Then

Answer 20

a) The set of solutions involves exactly n − r parameters.
b) If r < n, the system has infinitely many solutions.
c) If r = n, the system has a unique solution.