#### Study sets matching "algebra linear systems"

#### Study sets matching "algebra linear systems"

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…

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…

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…

system of linear equations

solution of a system of linear equations

substitution method

system of linear inequalities

two or more linear equations with the same variables

the ordered pair that makes both equations true at the same t…

one equation is solved for one of the variables then the the…

two or more linear inequalities with the same variables

system of linear equations

two or more linear equations with the same variables

solution of a system of linear equations

the ordered pair that makes both equations true at the same t…

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…

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…

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 system

dependent system

elimination method

inconsistent system

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

a system of equations that has infinitely many solutions

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

a system of equations or inequalities that has no solution

consistent system

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

dependent system

a system of equations that has infinitely many solutions

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…

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…

System of linear equations

Solve graphically

Solution to a system of linear equations

Infinite solutions

2 or more linear equations

Use a graph to find the solution

The point at which 2 linear equations intersect.

The number of solutions a system has when the graphs of the e…

System of linear equations

2 or more linear equations

Solve graphically

Use a graph to find the solution

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…

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

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.

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…

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

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…

elimination method

equivalent equations

infinitely many solutions

no solution

solves a system of equations by eliminating a variable by add…

equations that have the same solution(s)

linear systems of equations that have an infinite number of s…

when a system of equations fo have an intersection

elimination method

solves a system of equations by eliminating a variable by add…

equivalent equations

equations that have the same solution(s)

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

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…

Suppose we have a final solution set o…

How are two linear systems equivalent?

How do you solve a linear system?

A matrix with size 3x4 is: ___________…

infinite

They share the same solution sets

Replace the l.s. with one that is equivalent but "simpler", t…

augumented matrix

Suppose we have a final solution set o…

infinite

How are two linear systems equivalent?

They share the same solution sets

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.

Common Solution

Elimination

Substitution

System of Linear Equation

x-y coordinate of the graphical intersection of the linear eq…

The process of solving a linear system of equations by adding…

The process of solving a linear system of equations when one…

One or more linear equations together in a problem

Common Solution

x-y coordinate of the graphical intersection of the linear eq…

Elimination

The process of solving a linear system of equations by adding…

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…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…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 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…

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…

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

Indepent System

Inconsistent Sytem

Dependent System

Consistent

The lines intercept at 1 point and have 1 unique solution

The lines are parallel have no solution.

There are two equations for the same line. They have an infin…

(Intercepts)

Indepent System

The lines intercept at 1 point and have 1 unique solution

Inconsistent Sytem

The lines are parallel have no solution.

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…

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…

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…

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…

Matix

Elements of the matrix

Submatrix

Square matrix

A rectangular array of numbers.

The numbers in the array

A matrix composed of a array of numbers taken from a larger m…

When there are equal numbers of rows as Columns

Matix

A rectangular array of numbers.

Elements of the matrix

The numbers in the array

3.1 Two equivalent vectors must have t…

3.1 The vectors (a,b) and (a,b,0) are…

3.1 If k is a scalar and v is a vector…

3.1 The vectors v+(u+w) and (w+v)+u ar…

F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)

F (일단 기본적으로 차원이 같아야 함.)

F (벡터가 평행하다는 것은 굳이 방향이 같을 필요는 없음. 방향은 반대더라도 됨.)

T

3.1 Two equivalent vectors must have t…

F (옮겨서 시점이 달라져도 같은 벡터로 취급한다.)

3.1 The vectors (a,b) and (a,b,0) are…

F (일단 기본적으로 차원이 같아야 함.)

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 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…

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…

Suppose we have a final solution set o…

How are two linear systems equivalent?

How do you solve a linear system?

A matrix with size 3x4 is: ___________…

infinite

They share the same solution sets

Replace the l.s. with one that is equivalent but "simpler", t…

augumented matrix of *

Suppose we have a final solution set o…

infinite

How are two linear systems equivalent?

They share the same solution sets

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.

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…

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.