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Quadrant

Solution of an equation in two variables

Graph of an equation in two variables

Linear equation

One of the four regions into which the x and y axes divide th…

An ordered pair that produces a true statement when the coord…

The set of points in a coordinate plane that represent all so…

An equation whose graph is a line.

Quadrant

One of the four regions into which the x and y axes divide th…

Solution of an equation in two variables

An ordered pair that produces a true statement when the coord…

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.…

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

(i,j)-entry of A

The diagonal entries of A

Main diagonal

Diagonal matrix

The entry in A in the ith row and the jth column.

a11, a22,a33...

The diagonal of A formed by the diagonal entries.

A square n x n matrix whose nondiagonal entries are zero.

(i,j)-entry of A

The entry in A in the ith row and the jth column.

The diagonal entries of A

a11, a22,a33...

linear equation

slope

y-intercept

x-intercept

An equation in two variables whose graph in a coordinate plan…

A ratio comparing the change in output over the change in inp…

The value of the output when the input is zero (0, y). The po…

The value of the input when the output is zero (x, 0). The po…

linear equation

An equation in two variables whose graph in a coordinate plan…

slope

A ratio comparing the change in output over the change in inp…

rate or rate of change

constant rate of change

constant rate of change

zero rate

measure like miles per hour or price per item

rate that stays the same throughout the table

graph of this rate is a straight line

graph of this rate is a horizontal line

rate or rate of change

measure like miles per hour or price per item

constant rate of change

rate that stays the same throughout the table

Equivalent Systems

consistent system

inconsistent system

Row Echelon Form Qualifications [REF]…

Have the same solution set

either one solution or infinitely many

no solution

1. All zero rows below nonzero rows... 2. Each leading entry of…

Equivalent Systems

Have the same solution set

consistent system

either one solution or infinitely many

Linear Equation

Coefficients

System of Linear Equations/Linear System

Solution

a1x1 + a2x2 +... + anxn=b

a1...an are real or complex numbers.

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

A list of numbers that makes each equation a true statement w…

Linear Equation

a1x1 + a2x2 +... + anxn=b

Coefficients

a1...an are real or complex numbers.

basic variable

existance

consistent

inconsistant

variable that corresponds to a pivot column in the coefficien…

there is at least one solution to the matrix

the system of equations has at least one solution

the system has no solutions

basic variable

variable that corresponds to a pivot column in the coefficien…

existance

there is at least one solution to the matrix

Linearly independent and linearly depe…

Linear dependence relation

The columns of matrix A are linearly i…

A set containing only one vector is li…

the equation A**x** = **0** has only the trivial solution.

that vector is not the zero vector.

Linearly independent and linearly depe…

Linear dependence relation

Linear Equation

Linear Coefficients

System of linear equations (linear sys…

Solution of a linear system

1. variables x1......xn that can be written in the form:... a1x1…

Can be real or complex numbers

Collection of one of more linear equations that contain x1...…

List (s1, s2.....,sn) of numbers that makes each equation a t…

Linear Equation

1. variables x1......xn that can be written in the form:... a1x1…

Linear Coefficients

Can be real or complex numbers

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…

The leading entry of the row is the:

Echelon Matrix Form

Reduced Echelon Matrix Form

Inconsistent/Consistent Matrices and E…

leftmost nonzero entry in that row

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

1. Same three reqs. as echelon form... 2. Leading entry is 1... 3.…

1. If there is a solution, there is a reduced echelon form to…

The leading entry of the row is the:

leftmost nonzero entry in that row

Echelon Matrix Form

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

Homogeneous Linear System

Trivial/Nontrivial Solutions

The homogeneous equation Ax=0 has a no…

A nontrivial solution x can have some…

1. A system of linear equations is said to be homogeneous if…

1. The zero solution is called the trivial solution... 2. The no…

the equation has at least one free variable.

not all of the entries are zero.

Homogeneous Linear System

1. A system of linear equations is said to be homogeneous if…

Trivial/Nontrivial Solutions

1. The zero solution is called the trivial solution... 2. The no…

coefficient

distributive property

equation

inequality

the numerical part of a term

for any numbers a, b, and c: a(b+c) = ab+bc

a mathematical sentence showing two expressions are equal

mathematical sentences having the symbols <,>

coefficient

the numerical part of a term

distributive property

for any numbers a, b, and c: a(b+c) = ab+bc

Every linear transformation is...

Let T: R^n -> R^m be a linear transfor…

Standard matrix for the linear transfo…

Standard matrix for the linear transfo…

a matrix transformation (within finite dimensional spaces).

standard basis

Every linear transformation is...

a matrix transformation (within finite dimensional spaces).

Let T: R^n -> R^m be a linear transfor…

Transformation/function/mapping /T

Domain

Codomain

Range

A rule that assigns to each vector **x** in R^n to a vector in…

The vector space R^n on which a transformation acts.

The vector space R^m that contains (but is no necessarily exh…

The set of all possible images T(**x**) of transformation T.

Transformation/function/mapping /T

A rule that assigns to each vector **x** in R^n to a vector in…

Domain

The vector space R^n on which a transformation acts.

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.…

75 kilometers

300 kilometers

0 < y < 350

The original line went uphill and the…

A car travels at the rate shown in the graph. How far will it…

How far will the car travel in 6 hours?

Let f(x) = 5x where x represents speed in miles per hour of a…

What happens to the line represented by the equation, ... y = 4…

75 kilometers

A car travels at the rate shown in the graph. How far will it…

300 kilometers

How far will the car travel in 6 hours?

Consistent

Inconsistent

linear equation

system of linear equations

Either has one solution or infinitely many solutions

no solutions

in the variables x1...xn is an equation that can be written i…

collection of one or more linear equations involving the same…

Consistent

Either has one solution or infinitely many solutions

Inconsistent

no solutions

consistent linear system

inconsistent linear system

free variable

vector

a linear system with at least one solution

a linear system with no solution

any variable in a linear system that is not a basic variable

a list of numbers; a matrix with only one column

consistent linear system

a linear system with at least one solution

inconsistent linear system

a linear system with no solution

If the augmented matrices of 2 linear…

Another way to ask if a system is cons…

Echelon Form (properties)

Reduced Echelon Form (properties)

the 2 have the same solution set

does at least 1 solution exist

All non zero rows above zero rows... each leading entry of a ro…

All properties from Echelon Form... leading entry is 1 in each…

If the augmented matrices of 2 linear…

the 2 have the same solution set

Another way to ask if a system is cons…

does at least 1 solution exist

The number c in cu is a _________.

R^2 is a _______.

Vectors in R^3 are ______ X _______ co…

Vectors in R^3 are represented in ____…

scalar

rectangular coordinate system in a plane.

3; 1

three

The number c in cu is a _________.

scalar

R^2 is a _______.

rectangular coordinate system in a plane.

Linear Combination

Can b be written as a linear combinati…

Span{}

Vector Equations

Given a set of vectors v1, v2, v3,...vp in Rn, and scalars c1…

The vector equation c1**v1 + c2**v2 + ... cp*vp = b can be writ…

The set of ALL linear combinations of the vectors.

c1**v1 + c2**v2 + ... cp*vp = b

Linear Combination

Given a set of vectors v1, v2, v3,...vp in Rn, and scalars c1…

Can b be written as a linear combinati…

The vector equation c1**v1 + c2**v2 + ... cp*vp = b can be writ…

Linearly independent

Linearly dependent

Linear Independence of Matrix Columns

Set of One Vector

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

The columns of a matrix A are linearly independent iff the eq…

1. A set of one vector is linearly independent iff v is NOT t…

Linearly independent

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

Linearly dependent

1. Indexed set of vectors {v1,...vp} in R(n) is said to be li…

Vector

R^(n)

R2

Equal R2 vectors

An ordered list of members

-R stands for the real numbers that appear as entries in the…

The set of all vectors with two entries

Two vectors in R2 are equal iff their corresponding entries a…

Vector

An ordered list of members

R^(n)

-R stands for the real numbers that appear as entries in the…

Definition of Linear Combination

Matrix Equation

Theorem 3

Solution of Ax=b

1. If A is an m x n matrix, with columns a1, ... an, and if x…

Ax = b

1. If A is an m x n matrix with columns a1,...,an, and if b i…

1. The equation Ax=b has a solution iff b is a linear combina…

Definition of Linear Combination

1. If A is an m x n matrix, with columns a1, ... an, and if x…

Matrix Equation

Ax = b

Vector

Problem with definitions (6th Grade Ex…

Infusing Tech into Math Article

Velocity

a. Math. A quantity having direction as well as magnitude, de…

The truth is that all definitions, by their very nature, tend…

1) the student teachers' attitudes about using technology in…

Knowing that a ship has a speed of 10 knots (nautical miles p…

Vector

a. Math. A quantity having direction as well as magnitude, de…

Problem with definitions (6th Grade Ex…

The truth is that all definitions, by their very nature, tend…

Vector

Length of Vector

Unit Vector

scalar multiple

A real n-vector is an ordered sequence on n real #s. The set…

||a||=√a1^2+...+an^2

vector of length 1

let x=[x1...xn] be a vector in rn and let c be any scalar. th…

Vector

A real n-vector is an ordered sequence on n real #s. The set…

Length of Vector

||a||=√a1^2+...+an^2

Matrix equation

The matrix equation A**x** = **b** has the…

The equation A**x** = b only has a solut…

"The columns of A span R^m" means that…

the vector equation x1**a1** + x2**a2** + ... xn**an** = b if A is…

every vector in R^m is a linear combination of **v1 ... vp**

Matrix equation

The matrix equation A**x** = **b** has the…

the vector equation x1**a1** + x2**a2** + ... xn**an** = b if A is…

linear equation

system of linear equations

solution

solution set

a1x1+a2x2+...+anxn=b

collection of linear equations

a list of numbers that make each equation a true statement

a set of all possible solutions

linear equation

a1x1+a2x2+...+anxn=b

system of linear equations

collection of linear equations

linear equation

system of linear equations

solution of a system

consistent system

in the variables x₁,x₂,x₃..... is an equation that can be wri…

is a collection of linear equations involving the same variab…

of linear equations is a list of numbers that is a solution o…

one or infinitely solutions

linear equation

in the variables x₁,x₂,x₃..... is an equation that can be wri…

system of linear equations

is a collection of linear equations involving the same variab…

A linear equation is...

A system of linear equations (a linear…

A solution to the linear system x_1, .…

Solving a linear system means...

variables x_1, x_2,...,x_n, any equation that can be written…

x_1,x_2,...,x_n a collection of linear equations that is the…

any ordered n-tuple (S_1,...,S_n) of real numbers if you subs…

to find the solution set, i.e. the set of all possible soluti…

A linear equation is...

variables x_1, x_2,...,x_n, any equation that can be written…

A system of linear equations (a linear…

x_1,x_2,...,x_n a collection of linear equations that is the…

Row-Reduced Echelon Form

Row Equivalence

Equivalence Relation

Partition

1. Must be in Row Echelon Form... 2. Every lead coefficient is 1…

Two matrices, A and B, are row equivalent if it is possible t…

A binary relation on a set that must be reflexive, symmetric,…

A grouping of a set's elements that follows 3 rules:... 1. does…

Row-Reduced Echelon Form

1. Must be in Row Echelon Form... 2. Every lead coefficient is 1…

Row Equivalence

Two matrices, A and B, are row equivalent if it is possible t…

Two linear systems of equations are eq…

The three elementary row operations are

We say that vector v is a linear combi…

A function f: X->Y is onto if

they have the same solution set.

Replace one row by the sum of itself and a multiple of anothe…

there exist scalars x1, x2,..., xn satisfying v = x1v1+x2v2+.…

for each y ∈ Y there is at least one x ∈ X with f(x) = y

Two linear systems of equations are eq…

they have the same solution set.

The three elementary row operations are

Replace one row by the sum of itself and a multiple of anothe…

8 Vector space properties

Vector space definition

Section 1.2 thms (8 properties related)

Subspace definition

1. Commutativity: u + v = v + u for all u, v ∈ V ;... 2. Associa…

V is a vs iff:... ∃! x+y ∈ V for every x,y ∈ V... ∃! cv ∈ V for eve…

1. x+y = y+x... => x=z... 2. i) 0 vector is unique... ii) inverse vec…

x+y ∈W... cx∈W... 0vector∈W OR W is nonempty

8 Vector space properties

1. Commutativity: u + v = v + u for all u, v ∈ V ;... 2. Associa…

Vector space definition

V is a vs iff:... ∃! x+y ∈ V for every x,y ∈ V... ∃! cv ∈ V for eve…

consistent

inconsistent

row equivalent

Echelon form

when the system of equations has at least one solution

when the system of equations has no solutions

the systems have exactly the same solution set; Two matrices…

pivots can be any number; every all zero row of M is below ev…

consistent

when the system of equations has at least one solution

inconsistent

when the system of equations has no solutions