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1
Q

Homogeneous System of Equations

Definition

A

each of the equations in the system is equal to 0

2
Q

What are the three possibilities that can arise when we solve systems of linear equations?

A
  1. there is a unique solution
  2. there is no solution
  3. there are an infinite number of solutions
3
Q

Inconsistent

Definition

A

a system of linear equations are inconsistent if they have no solution

4
Q

Consistent

Definition

A

a system of linear equations are consistent if they have at least one solution

5
Q

Augmented Matrix

Definition

A

matrix composed of the coefficient matrix of the system of equations and the mx1 matrix b

6
Q

Row Operation

Definition

A

replaces a row with some function of that row

7
Q

Elementary Row Operation

Definition

A

One of the following:

i) swapping any two rows
ii) the multiplying of one row by another non-zero real number
iii) the adding or subtracting of a multiple of one row from another
- all elementary row operations are reversible

8
Q

Row Echelon Form

Definition

A

An mxn matrix is in row echelon form (REF) if:

1) if there are any zero rows, then they are at the bottom of the matrix
2) the first non-zero entry in each non-zero row is a 1, called a leading 1
3) in any non-zero row, the leading 1 is further to the right than the leading 1 in any previous row

9
Q

Reduced Row Echelon Form

Definition

A
  • the matrix satisfied all three conditions for a matrix in reduced row echelon form as well as the following condition:
    4) in each column that contains a leading 1, every other entry is zero
10
Q

The Matrix Method of Solving Systems of Linear Equations

A

1) replace the equations by the corresponding augmented matrix
2) apply elementary row operations in order to change the original to one in row echelon form (REF)
3) apply elementary row operations to change the matrix into reduced row echelon form (RREF)
4) read off the solutions (or lack of one) from the echelon form

11
Q

Free and Dependent Variables From RREF

A
  • columns without leading ones correspond to the free variables
  • columns with leading ones correspond to the dependent variables
  • the difference between the number of linearly independent equations in the system and the number of leading ones is equal to the number of free variables