Systems of Linear Equations

Question 1

Linear Equation

Answer 1

A linear equation in the 𝑛 variables 𝑥1,𝑥2,…,𝑥𝑛 is any equation that can be expressed in the form

𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎𝑛𝑥𝑛=𝑏,

where 𝑎₁,𝑎₂,…,𝑎𝑛 and 𝑏 belong to ℝ.

A system of linear equations (or simply a linear system) is a finite collection of linear equations in a fixed set of variables 𝑥1,𝑥2,…,𝑥𝑛.

Question 2

𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎𝑛𝑥𝑛=𝑏

what is a called

Answer 2

We call the number 𝑎ᵢ the coefficient of the variable 𝑥ᵢ

Question 3

𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎𝑛𝑥𝑛=𝑏

what is b called

Answer 3

𝑏 is the constant term of the equation

Question 4

What are some rules for what is and isn’t considered a linear equation?

Answer 4

A linear equation does not involve products of the variables like 𝑥1𝑥2, or powers of those variables (e.g. 𝑥31), or any trigonometric, exponential or logarithmic functions.

Question 5

homogeneous

Answer 5

When the constant term 𝑏 of a linear equation is zero, we call the linear equation homogeneous.
A linear system is called homogeneous if all constant terms are 0.

Question 6

s is what?

Answer 6

A vector 𝐬=[𝑠₁,𝑠₂,…,𝑠𝑛] is called a solution of the linear equation 𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎𝑛𝑥𝑛=𝑏 if it satisfies the equation, that is, if

𝑎₁𝑠₁+𝑎₂𝑠₂+⋯+𝑎𝑛𝑠𝑛=𝑏.

A solution of a system of linear equations is a vector which is simultaneously a solution of every equation in the system.

Question 7

What does the solution(s) of a linear equation in R², R³ and Rn form?

Answer 7

The set of all solutions of a two variable linear equation 𝑎𝑥+𝑏𝑦=𝑐, where at least one of 𝑎 or 𝑏 is non-zero, forms a line in ℝ2, and the set of all solutions of a three variable linear equation 𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑, where at least one of 𝑎 , 𝑏, or 𝑐 is non-zero, forms a plane in ℝ3. The set of all solutions to a linear equation in 𝑛 variables with at least one non-zero coefficient forms a hyperplane in ℝ𝑛.

Question 8

What is another way 𝑎₁𝑥₁+𝑎₂𝑥₂+⋯+𝑎𝑛𝑥𝑛=𝑏 can be expressed?

Answer 8

𝑎1𝑥1+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛=𝐚⋅𝐬

for 𝐚=[𝑎1,𝑎2,…𝑎𝑛] and 𝐱=[𝑥1,𝑥2,…𝑥𝑛]. So the linear equation 𝑎1𝑥1+𝑎2𝑥2+⋯+𝑎𝑛𝑥𝑛=𝑏 can be expressed as 𝐚⋅𝐱=𝑏. In the homogeneous case, we have 𝐚⋅𝐱=0, so 𝐬 is a solution of this homogeneous equation if and only if 𝐚⊥𝐬, that is, exactly when the vectors 𝐚 and 𝐬 are orthogonal.

Question 9

how can we solve a system of linear equations?

Answer 9

The linear systems which are easiest to solve are those systems whose augmented matrix is in reduced row echelon form (RREF). Since elementary row operations do not change the solution set of a system of linear equations, our strategy to solve a system of linear equations is to row reduce its augmented matrix [𝐴|𝐛] to reduced row echelon form and then solve the resulting system.

Question 10

consistent

Answer 10

A system of linear equations is called consistent if it has at least one solution.

Question 11

inconsistent

Answer 11

A system of linear equations is called inconsistent if it has no solutions.

Question 12

trivial solution

Answer 12

A homogeneous system of linear equations is always consistent since the zero vector
is always a solution of the system. We call this the trivial solution.

Question 13

solution set

Answer 13

The set of all solutions of a system of linear equations is called the solution set of the system.

Question 14

How many solutions is it possible for a system of linear equations to have?

Answer 14

A system of linear equations has either

a) a unique solution (i.e., exactly one solution),
b) infinitely many solutions, or
c) no solutions.

Question 15

the coefficient matrix

Answer 15

the coefficient matrix
when given a system of linear equations, it can be represented as a matrix where each column represents a variable in order from x₁ -> xn

Question 16

the augmented matrix

Answer 16

the same as the coefficient matrix, but now the last column represents the constant b of the linear equations

Question 17

row equivalent

Answer 17

Matrices A and B are said to be row equivalent if there exists a sequence of elementary row operations that converts A into B.

Question 18

What is the main theorem used to solve systems of linear equations?

Answer 18

Linear systems with row equivalent augmented matrices have the same solution set.

Question 19

leading variable

Answer 19

The variable 𝑥𝑖 is called a leading variable (or a basic variable) if the 𝑖th column of a row echelon form of [𝐴|𝐛] contains a leading entry.

Question 20

free variable

Answer 20

The variable 𝑥𝑖 is called a free variable if it is not a leading variable, that is, if the 𝑖th column of a row echelon form of [𝐴|𝐛] does not contain a leading entry.

Question 21

Gauss-Jordan elimination

Answer 21

There are three main steps to solve a system of linear equations:

1. Write the augmented matrix [𝐴|𝐛] of the system.
2. Use elementary row operations to find the reduced row echelon form of the augmented matrix [𝐴|𝐛].
3. If the resulting system is consistent, assign parameters to the free variables (if any). Solve for the leading variables in terms of those parameters (if any).

Question 22

Rank (A) is determined by

Answer 22

counting the number of leading variables in (A)
for Rank (A|B) count the leading variables in the entire augmented matrix

Question 23

When do you use parameters in a system of linear equations?

Answer 23

For any free variables, you can solve for the leading variables and set free variables to s,t etc as parameters

Question 24

how can we tell if a system is inconsistent?

Answer 24

A linear system is inconsistent if and only if the last non-zero row in a row echelon form of the augmented matrix [𝐴|𝐛] has the form

[00⋯0|𝑐]

for some non-zero 𝑐∈ℝ, that is, there is a leading entry in the last column of a row echelon form [𝐴|𝐛]. Thus, a linear system is inconsistent if and only if
rank([𝐴|𝐛])=rank(𝐴)+ 1

Question 25

how is rank affected by a system being inconsistent?

Answer 25

rank([𝐴|𝐛])={rank(𝐴) ⟺ the system is consistent

rank(𝐴)+1 ⟺ the system is inconsistent

Question 26

how is rank affected by a system having a unique solution?

Answer 26

For a system to have a unique solution, it cannot have any free variables! Every variable must be a leading variable. From the Rank Theorem, we know that for a system to have a unique solution, rank(𝐴)=𝑛 ( = # of columns of 𝐴 ).

Question 27

for a consistent system, how does the number of free variables affect the solution(s)?

Answer 27

For any consistent system, the number of free variables is equal to the number of parameters )s,t) in the general solution of the system.