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Study sets matching "algebra linear systems"

12 terms
Linear Algebra: Systems of equations
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…
8 terms
Algebra I - Systems of Linear Equations
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…
7 terms
Algebra 1: Systems of Linear Equations
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…
12 terms
Linear Algebra: Systems of equations
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…
7 terms
Linear Systems - Algebra II
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…
9 terms
Algebra linear systems
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…
Algebra; linear systems
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…
17 terms
Linear Systems of Equations (Algebra 1)
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…
6 terms
Algebra - Systems of Linear equations
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
9 terms
THE SHAPES OF ALGEBRA LINEAR SYSTEMS AND INEQUALITIES
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…
12 terms
Apex Algebra Unit 8: Systems of Linear Equations
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…
9 terms
Algebra lesson 11 ( Linear systems)
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.
10 terms
Algebra 2 Unit 3 Linear Systems
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
16 terms
Thinkwell Algebra 2 - Ch 3 Linear Systems
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…
13 terms
Algebra 1-Unit 6: Systems of Linear Equations
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…
126 terms
Linear Algebra
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…
10 terms
Algebra 1: Unit 4 Systems of Linear Equations & Inequalities
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
15 terms
Linear Algebra
Slope Intercept Form
System of linear equations
Point Slope Form
The answer to a system of linear equat…
y= mx + b
2x + y = 5 ... 3x - y = 6
y - y^1 = m(x - x^1)
Ordered Pair
Slope Intercept Form
y= mx + b
System of linear equations
2x + y = 5 ... 3x - y = 6
66 terms
Linear Algebra
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 en, [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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
126 terms
Linear Algebra
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…
66 terms
Linear Algebra
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 en, [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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
67 terms
Linear algebra
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.
63 terms
Algebra II - Function notations and Linear Systems
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…
30 terms
Honors Algebra 2 Unit 3 - Linear Systems and Matrices
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…
67 terms
Linear algebra
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.
67 terms
Linear algebra
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.
Linear Algebra Section 1.5: Solutions of linear systems
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.
Linear Algebra 1.5: Solution Sets of Linear Systems
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.
9 terms
Algebra - Unit 5 Systems of Linear Equations and Inequalties
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…
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
34 terms
Linear Algebra
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
8 terms
Algebra I - Systems of Linear Equations and Inequalities set 2
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…
16 terms
Algebra1 (Monarch) Unit 5 Linear Systems
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…
12 terms
Algebra 1 Systems of Linear Equations and Inequalities
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…
31 terms
Algebra 1 Linear Review
D
A
D
B
D
A
9 terms
Algebra Linear functions
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…
92 terms
linear algebra
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…
30 terms
Algebra function; Graph, Function, and linear system
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…
8 terms
Algebra II Ch.3: Linear Systems
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…
Linear Algebra Chapter 1
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
66 terms
Linear Algebra
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 en, [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 en, [1, 0, 0, 0...0] to the nth number. So in R3…
11 terms
11-algebra unit 4 linear systems
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…
100 terms
Linear Algebra
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…
42 terms
Linear Algebra
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…
32 terms
Linear Algebra Test 1
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.…
51 terms
Linear Algebra
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…
97 terms
Linear algebra
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…
42 terms
Linear Algebra
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.
15 terms
Linear Algebra
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…
65 terms
Linear Algebra
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…
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