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Graphically, a system of equations has…

Graphically, a system of equations has…

The three possible solutions for a sys…

A system has infinite solutions if:

They are parallel lines. They never intersect so there's no w…

at the point of intersection. The point where lines have a co…

- no solution... - infinite solutions ... - a unique solution

The equations are equal.... Same slope and y-intercept.

Graphically, a system of equations has…

They are parallel lines. They never intersect so there's no w…

Graphically, a system of equations has…

at the point of intersection. The point where lines have a co…

consistent

dependent

elimination

inconsistent

a system of linear equations that has at least one ordered pa…

a system of linear equations that has an infinite number of s…

the use of addition or subtraction to eliminate one variable…

a system of equations with no ordered pair that satisfies bot…

consistent

a system of linear equations that has at least one ordered pa…

dependent

a system of linear equations that has an infinite number of s…

elimination method

linear inequality

solution of a system of linear inequal…

solution of a system of linear equations

solving systems by adding or subtracting equations to elimina…

an inequality in two variables whose graph is a region of the…

a set of values that make all inequalities in a system true.…

a set of values that make all equations in a system true. [p.…

elimination method

solving systems by adding or subtracting equations to elimina…

linear inequality

an inequality in two variables whose graph is a region of the…

Graphically, a system of equations has…

Graphically, a system of equations has…

The three possible solutions for a sys…

A system has infinite solutions if:

They are parallel lines. They never intersect so there's no w…

at the point of intersection. The point where lines have a co…

- no solution... - infinite solutions ... - a unique solution

The equations are equal.... Same slope and y-intercept.

Graphically, a system of equations has…

They are parallel lines. They never intersect so there's no w…

Graphically, a system of equations has…

at the point of intersection. The point where lines have a co…

Optomizing

Steps For Optomizing

No Solution

Infinitely Many Solutions

Maximize profit, minimize cost

1. Define variables (always first)... 2. Write system of inequal…

Inconsistent

Consistent, Dependent

Optomizing

Maximize profit, minimize cost

Steps For Optomizing

1. Define variables (always first)... 2. Write system of inequal…

System of linear equations

Point of intersection

Solution of a linear system

Substitution method

Two or more linear equations in the same variables, x and y.

The ordered pair (a, b) that lies on the graphs of two or mor…

An ordered pair (x, y) that makes each equation in the system…

Solving for one variable and substituting the result in the s…

System of linear equations

Two or more linear equations in the same variables, x and y.

Point of intersection

The ordered pair (a, b) that lies on the graphs of two or mor…

System of linear equations

Point of intersection

Solution of a linear system

Substitution method

Two or more linear equations in the same variables (x,y)

Ordered pair (a,b) that lies on the graphs of two or more equ…

Solving for one variable and substituting the result in the s…

Solving for one variable and substituting the result

System of linear equations

Two or more linear equations in the same variables (x,y)

Point of intersection

Ordered pair (a,b) that lies on the graphs of two or more equ…

Function

Function Notation

Input

Output

A relationship where each input has exactly one output.

If x is input and y is the output, then the function notation…

A number in the domain of a function. ... The independent variab…

A number in the range of a function.... The dependent variable.

Function

A relationship where each input has exactly one output.

Function Notation

If x is input and y is the output, then the function notation…

In a system of equations with n distin…

Two ways to solve a system of linear e…

What is Substitution

What is Combination

You need n number of linear equations

Substitution... Combination

Solve one equation in terms of one varible and substitue that…

Add or Substract one equation from the other to cancel out on…

In a system of equations with n distin…

You need n number of linear equations

Two ways to solve a system of linear e…

Substitution... Combination

CHORD

CIRCLE

LINEAR EQUATION IN STANDARD FORM

MIDPOINT

A LINE SEGMENTWITH END POINTS ON A CIRCLE.

A GEOMETRIC FIGURE CONSISTING OF ALL POINTS P THAT ARE FIXED…

THE FORM AX+BY=C OF A LINEAR EQUATION.

A POINT ON A LINE SEGMENT THAT IS EDUIDISTANT FROM THE END PO…

CHORD

A LINE SEGMENTWITH END POINTS ON A CIRCLE.

CIRCLE

A GEOMETRIC FIGURE CONSISTING OF ALL POINTS P THAT ARE FIXED…

Linear Equation

Ordered Pairs

Slope

Slope-Intercept Form

an equation whose solutions form a straight line on a coordin…

a pair of numbers that can be used to locate a point on a coo…

a measure of the steepness of a line on a graph; calculated a…

a linear equation written in the form y = mx + b, where m rep…

Linear Equation

an equation whose solutions form a straight line on a coordin…

Ordered Pairs

a pair of numbers that can be used to locate a point on a coo…

consistent system

dependent system

inconsistent system

independent system

A system of equation or inequalities that has at least one so…

a system of equations that has infinitely many solutions.

a system of equations or inequalities that has no solution.

a system of equations that exactly one solution.

consistent system

A system of equation or inequalities that has at least one so…

dependent system

a system of equations that has infinitely many solutions.

System of Equations

Linear System

Solution of a System

Consistent System

a set of 2 or more equations

a system with linear equations

the set of values of the variable that makes all of the equat…

a system with at least one solution

System of Equations

a set of 2 or more equations

Linear System

a system with linear equations

consistent system

elimination

independent system

objective function

a system of equations or inequalities that has at least one s…

a method used to solve systems of equations in which one vari…

a system of equations that has exactly one solution

the function to be maximized or minimized in a linear program…

consistent system

a system of equations or inequalities that has at least one s…

elimination

a method used to solve systems of equations in which one vari…

elimination method

linear inequality

solution of a system of linear inequal…

solution of a system of linear equations

solving systems by adding or subtracting equations to elimina…

an inequality in two variables whose graph is a region of the…

a set of values that make all inequalities in a system true.

a set of values that make all equations in a system true.

elimination method

solving systems by adding or subtracting equations to elimina…

linear inequality

an inequality in two variables whose graph is a region of the…

Reduced row echelon form conditions

Theorem "dimension characterizes isomo…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an…

1. leading entry (if there is one) is 1 2. if column contains…

For subspaces V within R^n and W within R^m, we have V is iso…

There exists an isomorphism T:W->V

"isomorphic to" is an equivalence relation for subspaces V wi…

Reduced row echelon form conditions

1. leading entry (if there is one) is 1 2. if column contains…

Theorem "dimension characterizes isomo…

For subspaces V within R^n and W within R^m, we have V is iso…

System of Equations

System of Inequalities

Consistent

Inconsistent

A set of equations with the same variables.

A set of two or more inequalities with the same variables

A system of equations that has at least one solution is consi…

A system of equations that has no solution.

System of Equations

A set of equations with the same variables.

System of Inequalities

A set of two or more inequalities with the same variables

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Reduced row echelon form conditions

Theorem "dimension characterizes isomo…

Define "W is isomorphic to V"

Theorem stating "isomorphic to" is an…

1. leading entry (if there is one) is 1 2. if column contains…

For subspaces V within R^n and W within R^m, we have V is iso…

There exists an isomorphism T:W->V

"isomorphic to" is an equivalence relation for subspaces V wi…

Reduced row echelon form conditions

1. leading entry (if there is one) is 1 2. if column contains…

Theorem "dimension characterizes isomo…

For subspaces V within R^n and W within R^m, we have V is iso…

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

1

2

3

5

A table represents a function when the input values are uniqu…

A table represents a function when the input values are not u…

Domain is the input and Range is the output *(true = 3 or fal…

Domain is the output and Range is the input*(true = 4 or fals…

1

A table represents a function when the input values are uniqu…

2

A table represents a function when the input values are not u…

System of Two Linear Equations

Solution

Consistent System

Inconsistent Solution

Consists of two equations that can be written as:... Ax + By = C…

An ordered pair (x,y) that satisfies each equation in the sys…

A system that has at least one solution.

A system that has no solutions.

System of Two Linear Equations

Consists of two equations that can be written as:... Ax + By = C…

Solution

An ordered pair (x,y) that satisfies each equation in the sys…

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

What is a subspace and what are the re…

Column Space?

What is closure?

What is true for the span of a set of…

A subset (H) of a larger set of vectors(V) that has 3 propert…

denoted col(A)... the span of the columns of A.

When an operation such as addition is done with two numbers f…

For a set of vectors {v₁, v₂, v₃....} in Rⁿ... span{v₁,v₂,v₃...…

What is a subspace and what are the re…

A subset (H) of a larger set of vectors(V) that has 3 propert…

Column Space?

denoted col(A)... the span of the columns of A.

When Ax=b has no solution,

Linear system is homogenous if

Is a homogenous solution consistent or…

Trivial solution

the solution set is empty

it is of the form Ax = 0 vector in matrix equation form.

ALWAYS consistent because you can always write down the solut…

x = 0; of Ax = 0

When Ax=b has no solution,

the solution set is empty

Linear system is homogenous if

it is of the form Ax = 0 vector in matrix equation form.

Homogeneous Solution:

Trivial Solution:

Nontrivial Solution:

[THEOREM] The homogeneous equation Ax…

Ax = 0; 0 is the zero vector.

the zero solution.

a nonzero vector x that satisfies Ax = 0.

the equation has at least one free variable.

Homogeneous Solution:

Ax = 0; 0 is the zero vector.

Trivial Solution:

the zero solution.

Systems of Linear Equations

Systems of Linear Equalities

Solution to a System of Linear Equations

Solution to a System of Linear Inequal…

set or collection of equations that you deal with all togethe…

inequalities in two variables consists of at least two linear…

must satisfy all of the equations (solution set is the inters…

the ordered pair that is a solution to all inequalities in th…

Systems of Linear Equations

set or collection of equations that you deal with all togethe…

Systems of Linear Equalities

inequalities in two variables consists of at least two linear…

linear equation

consistent system

inconsistent system

leading entry

An equation that can be written as a1x1 + a2x2 + ... = b; a1,…

Has one or infinitely many solutions

Has no solution

Leftmost non-zero entry in a non-zero row

linear equation

An equation that can be written as a1x1 + a2x2 + ... = b; a1,…

consistent system

Has one or infinitely many solutions

consistent

dependent

elimination

inconsistent

a system of linear equations that has at least one ordered pa…

a system of linear equations that has an infinite number of s…

the use of addition or subtraction to eliminate one variable…

a system of equations with no ordered pair that satisfies bot…

consistent

a system of linear equations that has at least one ordered pa…

dependent

a system of linear equations that has an infinite number of s…

coefficient

consistent equations

dependent equations

determinant

the constant preceding the variables in a product

A system of linear equations that contain at least one common…

A system of linear equations that rely on each other for the…

the value of: (row 1, column 1)(row 2 column 2) - (row 1, col…

coefficient

the constant preceding the variables in a product

consistent equations

A system of linear equations that contain at least one common…

system of linear equations

solution of a system

elimination method

substitution method

Two or more linear equations using the same variables.

Any ordered pair (x,y) that makes both equations true in a sy…

A method for solving systems by adding or subtracting equatio…

A method for solving systems by replacing one variable with a…

system of linear equations

Two or more linear equations using the same variables.

solution of a system

Any ordered pair (x,y) that makes both equations true in a sy…

Y- intercept

Slope

Slope- intercept Form

Point- slope form

the y-coordinate of a point where a graph crosses the y-axis

the steepness of a line on a graph, equal to its vertical cha…

an equation written in the form y=mx+b is in slope-intercept…

y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point…

Y- intercept

the y-coordinate of a point where a graph crosses the y-axis

Slope

the steepness of a line on a graph, equal to its vertical cha…

matrix notation for an entry

augmented matrix

row-echelon form

Gaussian elimination

(A)ij means the entry for A in the i row and the j column. Or…

where you have the solutions set up as another column; the ri…

Any rows entirely of 0 are at the bottom. Each row's first no…

*I think* where you do row operations starting at the top and…

matrix notation for an entry

(A)ij means the entry for A in the i row and the j column. Or…

augmented matrix

where you have the solutions set up as another column; the ri…

Linear equation

Standard form

Constant

Table method

...... An equation whose graph is a line

...... Ax + By=C, where A, B, and C are not decimals or fraction…

...no... A value that does not change

...... 1. Number of valence electron

Linear equation

...... An equation whose graph is a line

Standard form

...... Ax + By=C, where A, B, and C are not decimals or fraction…

System of Equations

Linear system

Solution of a System

Inconsistent system

Is a set of two or more equations using the same variables

Is a set of tow or more linear equations that use the same va…

A solution of a system is a set of values for the variables t…

A system of equations that has no solution. It has No solutio…

System of Equations

Is a set of two or more equations using the same variables

Linear system

Is a set of tow or more linear equations that use the same va…

The number of solutions in a system of…

A system is said to be CONSISTENT if...

A system is said to be INCONSISTENT if…

Echelon Form

None, One, or Infinitely Many

it has one or infinitely many solutions

it has no solution

1- rows of zeroes on the bottom, 2- each leading entry of a r…

The number of solutions in a system of…

None, One, or Infinitely Many

A system is said to be CONSISTENT if...

it has one or infinitely many solutions

Scalar multiplication

Standard Basis in Rn

Closed under addition

Closed under scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Two vectors, when added together, are still in the same subsp…

When a vector is multiplied by a scalar, it's still in the sa…

Scalar multiplication

Multiplication of vectors by t, such that it becomes t(v1), t…

Standard Basis in Rn

Denoted by **e**n, [1, 0, 0, 0...0] to the nth number. So in R3…

Consistent system

Dependent system

Inconsistent system

Independent system

A system of one linear equations is consistent if it has at l…

A system of equations that does not have a unique solution is…

A system of equations that has no solution is this. Is a syst…

A system of linear equations that has a unique solution is th…

Consistent system

A system of one linear equations is consistent if it has at l…

Dependent system

A system of equations that does not have a unique solution is…

linear equation

system of linear equations

solution set

equivalent linear systems

an equation that can be written in the form a1x1 + a2x2 + ...…

a collection of one or more linear equations involving the sa…

a list of numbers that makes each equation in a system a true…

linear systems with the same solution set

linear equation

an equation that can be written in the form a1x1 + a2x2 + ...…

system of linear equations

a collection of one or more linear equations involving the sa…

Elementary Row Operations

Row Equivalency

Two equivalent equivalent augmented ma…

Two Fundamental Questions

1. Replacement... 2. Scaling... 3. Interchange

A is row equivalent to B if we can get from A to B via a sequ…

solution set

1. Is the system consistent... 2. If the solution exists, is the…

Elementary Row Operations

1. Replacement... 2. Scaling... 3. Interchange

Row Equivalency

A is row equivalent to B if we can get from A to B via a sequ…

Vector Space

Vector Space Axioms

Subspace

Transpose

a set where addition and scalar multiplication as well as the…

1. Commutativity of addition... 2. Associativity in addition... 3.…

a subset of a vector space that is also a vector space closed…

Matrix formed from switching rows and colums (A₁₂)'=A₂₁

Vector Space

a set where addition and scalar multiplication as well as the…

Vector Space Axioms

1. Commutativity of addition... 2. Associativity in addition... 3.…

Linear combination

Linearly independent

Theorem 5.1.1

elementary vectors

an expression of the form c1v1+c2v2+...+cnvn where all the ci…

a set of vectors in LI if the only way to write the zero vect…

A set of vectors is LD iff one of the vectors can be written…

vectors that have exactly one component equal to 1 and all ot…

Linear combination

an expression of the form c1v1+c2v2+...+cnvn where all the ci…

Linearly independent

a set of vectors in LI if the only way to write the zero vect…

Definition of a vector space

8 axioms of a vector space

Notation for a vector space with n dim…

Definition of a linear combination of…

V (set of elements: x, y, u, v) is called a vector space of K…

1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…

ℝⁿ

*x* is a linear combination of *v₁, ... vₙ* if for *v₁, ... v…

Definition of a vector space

V (set of elements: x, y, u, v) is called a vector space of K…

8 axioms of a vector space

1) *u, v, w* ∈ V, (*u* + *v*) + *w* = *u* + (*v* + *w*) (Asso…

Solution of an SLE

Consistent SLE

Inconsistent SLE

Equivalent linear system

A SLE has (1) no solution, or (2) exactly one solution, or (3…

SLE has one or infinitely many solutions.

SLE has no solution.

A SLE that have the same solution set

Solution of an SLE

A SLE has (1) no solution, or (2) exactly one solution, or (3…

Consistent SLE

SLE has one or infinitely many solutions.

Elementary Matrix

property of Elementary Matrix

Cramer's Rule2

subspace of Rn

what an n x n matrix E is called if we can obtain E from I, b…

each of these elementary row operations can be implemented by…

(a) the sum of any pair of vectors in the set lies in the set…

Elementary Matrix

what an n x n matrix E is called if we can obtain E from I, b…

property of Elementary Matrix

each of these elementary row operations can be implemented by…

Solving if something is in a Span

Span Theorems

Span

Subset

By definition of span, we want to know if there are scalars x…

Theorem 1: The subspace spanned by a non-empty subset S of a…

The set of all linear combinations of the vectors.... or ... The…

A subset is a set of vectors. Assume a subset V∈ℜn... Can be a…

Solving if something is in a Span

By definition of span, we want to know if there are scalars x…

Span Theorems

Theorem 1: The subspace spanned by a non-empty subset S of a…