Systems of Linear Equations Flashcards
Linear Equation
A linear equation in the π variables π₯1,π₯2,β¦,π₯π is any equation that can be expressed in the form
πβπ₯β+πβπ₯β+β―+πππ₯π=π,
where πβ,πβ,β¦,ππ and π belong to β.
A system of linear equations (or simply a linear system) is a finite collection of linear equations in a fixed set of variables π₯1,π₯2,β¦,π₯π.
πβπ₯β+πβπ₯β+β―+πππ₯π=π
what is a called
We call the number πα΅’ the coefficient of the variable π₯α΅’
πβπ₯β+πβπ₯β+β―+πππ₯π=π
what is b called
π is the constant term of the equation
What are some rules for what is and isnβt considered a linear equation?
A linear equation does not involve products of the variables like π₯1π₯2, or powers of those variables (e.g. π₯31), or any trigonometric, exponential or logarithmic functions.
homogeneous
When the constant term π of a linear equation is zero, we call the linear equation homogeneous.
A linear system is called homogeneous if all constant terms are 0.
s is what?
A vector π¬=[π β,π β,β¦,π π] is called a solution of the linear equation πβπ₯β+πβπ₯β+β―+πππ₯π=π if it satisfies the equation, that is, if
πβπ β+πβπ β+β―+πππ π=π.
A solution of a system of linear equations is a vector which is simultaneously a solution of every equation in the system.
What does the solution(s) of a linear equation in RΒ², RΒ³ and Rn form?
The set of all solutions of a two variable linear equation ππ₯+ππ¦=π, where at least one of π or π is non-zero, forms a line in β2, and the set of all solutions of a three variable linear equation ππ₯+ππ¦+ππ§=π, where at least one of π , π, or π is non-zero, forms a plane in β3. The set of all solutions to a linear equation in π variables with at least one non-zero coefficient forms a hyperplane in βπ.
What is another way πβπ₯β+πβπ₯β+β―+πππ₯π=π can be expressed?
π1π₯1+π2π₯2+β―+πππ₯π=πβ π¬
for π=[π1,π2,β¦ππ] and π±=[π₯1,π₯2,β¦π₯π]. So the linear equation π1π₯1+π2π₯2+β―+πππ₯π=π can be expressed as πβ π±=π. In the homogeneous case, we have πβ π±=0, so π¬ is a solution of this homogeneous equation if and only if πβ₯π¬, that is, exactly when the vectors π and π¬ are orthogonal.
how can we solve a system of linear equations?
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 [π΄|π] to reduced row echelon form and then solve the resulting system.
consistent
A system of linear equations is called consistent if it has at least one solution.
inconsistent
A system of linear equations is called inconsistent if it has no solutions.
trivial solution
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.
solution set
The set of all solutions of a system of linear equations is called the solution set of the system.
How many solutions is it possible for a system of linear equations to have?
A system of linear equations has either
a) a unique solution (i.e., exactly one solution),
b) infinitely many solutions, or
c) no solutions.
the coefficient matrix
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 xβ -> xn
the augmented matrix
the same as the coefficient matrix, but now the last column represents the constant b of the linear equations
row equivalent
Matrices A and B are said to be row equivalent if there exists a sequence of elementary row operations that converts A into B.
What is the main theorem used to solve systems of linear equations?
Linear systems with row equivalent augmented matrices have the same solution set.
leading variable
The variable π₯π is called a leading variable (or a basic variable) if the πth column of a row echelon form of [π΄|π] contains a leading entry.
free variable
The variable π₯π is called a free variable if it is not a leading variable, that is, if the πth column of a row echelon form of [π΄|π] does not contain a leading entry.
Gauss-Jordan elimination
There are three main steps to solve a system of linear equations:
- Write the augmented matrix [π΄|π] of the system.
- Use elementary row operations to find the reduced row echelon form of the augmented matrix [π΄|π].
- 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).
Rank (A) is determined by
counting the number of leading variables in (A) for Rank (A|B) count the leading variables in the entire augmented matrix
When do you use parameters in a system of linear equations?
For any free variables, you can solve for the leading variables and set free variables to s,t etc as parameters
how can we tell if a system is inconsistent?
A linear system is inconsistent if and only if the last non-zero row in a row echelon form of the augmented matrix [π΄|π] has the form
[00β―0|π]
for some non-zero πββ, that is, there is a leading entry in the last column of a row echelon form [π΄|π]. Thus, a linear system is inconsistent if and only if
rank([π΄|π])=rank(π΄)+ 1
how is rank affected by a system being inconsistent?
rank([π΄|π])={rank(π΄) βΊ the system is consistent
rank(π΄)+1 βΊ the system is inconsistent
how is rank affected by a system having a unique solution?
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(π΄)=π ( = # of columns of π΄ ).
for a consistent system, how does the number of free variables affect the solution(s)?
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.
