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discrete relation

continuous relation

vertical line test

independent variable

domain is a set of individual points

domain of relation has an infinite number of elements and the…

can be used to determine whether it's a function

the variable, often x, with values making up the domain

discrete relation

domain is a set of individual points

continuous relation

domain of relation has an infinite number of elements and the…

analytic geometry

boundary

composite

composition

prove a theorem from plane geometry by placing the figure in…

if an inequality is <= or >= the ________ is drawn a solid li…

the new function created by mapping one set onto another

combining two functions

analytic geometry

prove a theorem from plane geometry by placing the figure in…

boundary

if an inequality is <= or >= the ________ is drawn a solid li…

correlation

(quantitative) bi-variate data

causation

negative correlation

description of the linear relationship between the 2 variable…

data for TWO related variables that can represented by a scat…

a relationship between two variables in which one variable di…

trend descends L to R

correlation

description of the linear relationship between the 2 variable…

(quantitative) bi-variate data

data for TWO related variables that can represented by a scat…

one-to-one function

onto function

discrete relation

continuous relation

A function where each element of the range is paired with exa…

Each element of the range corresponds to an element of the do…

A relation in which the domain is a set of individual points.

A relation that can be graphed with a line or smooth curve.

one-to-one function

A function where each element of the range is paired with exa…

onto function

Each element of the range corresponds to an element of the do…

Mathematical

Variable

Constant Rate

Mathematical Variable

Having to do with numbers.

Something that can change like the height of a plant.

Stay at the same speed

Numbers that can change; symbolized with a letter. Example h…

Mathematical

Having to do with numbers.

Variable

Something that can change like the height of a plant.

Mathematical

Variable

Constant Rate

Mathematical Variable

Having to do with numbers.

Something that can change like the height of plant.

Stay at the same speed

Numbers that can change; symbolized with a letter. Example he…

Mathematical

Having to do with numbers.

Variable

Something that can change like the height of plant.

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…

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.

Variable

Mathematical Variable

Linear

Rise

Something that can change like the height of plant.

Numbers that can change; symbolized with a letter. Example he…

Relation that graphs a straight line.

How far "up" you go. Vertical Height

Variable

Something that can change like the height of plant.

Mathematical Variable

Numbers that can change; symbolized with a letter. Example he…

cartesian plane

coordinates

quadrants

origin

a plane divided into four quadrants by the intersection of th…

Ordered pairs that identify points on a coordinate plane. (x,y)

Four regions into which a coordinate plane is divided by the…

(0,0) The point of intersection of the x-axis and y-axis in a…

cartesian plane

a plane divided into four quadrants by the intersection of th…

coordinates

Ordered pairs that identify points on a coordinate plane. (x,y)

Mathematical

Variable

Constant Rate

Mathematical Variable

Having to do with numbers.

Something that can change like the height of a plant..

Stay at the same speed

Numbers that can change; symbolized with a letter. (Height =…

Mathematical

Having to do with numbers.

Variable

Something that can change like the height of a plant..

Variable

Constant Rate

Mathematical Variable

Linear

Something that can change like the height of plant.

Stay at the same speed

Numbers that can change; symbolized with a letter. Example he…

Relation that graphs a straight line.

Variable

Something that can change like the height of plant.

Constant Rate

Stay at the same speed

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.

cartesian plane

coordinates

origin

linear relation

a plane divided into four quadrants by the intersection of th…

Ordered pairs that identify points on a coordinate plane. (x,y)

(0,0) The point of intersection of the x-axis and y-axis in a…

A type of relationship that exists between two variables (usu…

cartesian plane

a plane divided into four quadrants by the intersection of th…

coordinates

Ordered pairs that identify points on a coordinate plane. (x,y)

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

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…

Direct Variation

Partial Variation

y = kx

y = kx + b

A line on a graph that goes through the origin (0,0).

A line on a graph that does not go through the origin (0,0).

The equation of a direct variation.

The equation of a partial variation.

Direct Variation

A line on a graph that goes through the origin (0,0).

Partial Variation

A line on a graph that does not go through the origin (0,0).

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…

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…

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…

Variable

Isolate The Variable

Solution

Greater Than

A symbol used to represent a quantity that can change

To get a variable alone on one side of an equation or inequal…

The set containing values of the variables

Used to compare two numbers when the first number is larger t…

Variable

A symbol used to represent a quantity that can change

Isolate The Variable

To get a variable alone on one side of an equation or inequal…

When factorising

Inverse

To solve linear equations

Cross multiplying

Take out the number in brackets

When you multiply something by its inverse you get one

Work backwards with inverse operations

Only when its equal two fraction... Basically lift and swap the…

When factorising

Take out the number in brackets

Inverse

When you multiply something by its inverse you get one

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…

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

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…

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.…

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…

the image of at least one x ∈ R.

T(u + v) = T(u) + T(v)

T(cu) = cT(u)

row equivalent

A transformation T : Rn → Rm is one-to-one if each b ∈ Rm is…

_________ for all u, v in the domain of T.

_________ for all scalars c and all vectors u in the domain o…

two matrices are _________ if one can be changed to the other…

the image of at least one x ∈ R.

A transformation T : Rn → Rm is one-to-one if each b ∈ Rm is…

T(u + v) = T(u) + T(v)

_________ for all u, v in the domain of T.

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.)

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

Truculent

Rapacious

Misanthrope

Craven

Eager or quick to argue or fight; aggressively defiant. ... Syn:…

Person who is abnormally anxious about their health.... Syn: Val…

General hatred, distrust and contempt of the human species or…

A cowardly person; contemptibly lacking in courage. "A craven…

Truculent

Eager or quick to argue or fight; aggressively defiant. ... Syn:…

Rapacious

Person who is abnormally anxious about their health.... Syn: Val…

Linear Equation

System of Linear Equations

Solution

Solution Set

A LINEAR EQUATION in variables x₁,x₂,....,xⁿ (for n a positiv…

A SYSTEM OF LINEAR EQUATIONS is a collection of one or more l…

A SOLUTION to a linear system is a list (s₁,s₂,...,sⁿ) of num…

The set of all solutions to a linear system is called the SOL…

Linear Equation

A LINEAR EQUATION in variables x₁,x₂,....,xⁿ (for n a positiv…

System of Linear Equations

A SYSTEM OF LINEAR EQUATIONS is a collection of one or more l…

What is a subspace and what are the re…

Column Space?

column rank

Row space

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

denoted col(A)... the span of the columns of A.... the basis are th…

cr:=dim(column space)

denoted row(A)... the span of rows of A... the basis is all nonzero…

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.... the basis are th…

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

A finite basis for V of Field F is a f…

Let S be a nonempty subset of a vector…

Let V & W be vector spaces over field…

A nonempty finite subset {u1, u2,.....…

Finite Basis

Span

Linear Transformations (Operators)

Linear Dependent

A finite basis for V of Field F is a f…

Finite Basis

Let S be a nonempty subset of a vector…

Span

Order of Operations

Inequality

Like Terms

Equivalent expressions

Ch. 1 PEMDAS the order in which you solve an equation. (Paren…

Ch. 1 An open sentence that contains the symbols < >

Ch. 1 Terms that contain the same variables, with correspondi…

Ch. 1 Expressions that simplify to the same number.

Order of Operations

Ch. 1 PEMDAS the order in which you solve an equation. (Paren…

Inequality

Ch. 1 An open sentence that contains the symbols < >

What is the null space of matrix A? Wh…

A subspace of Rⁿ is any set H in Rⁿ th…

What is the column space of matrix A?

The set {e₁; ...; eη} is called the __…

The null space of a matrix A is the set Nul A of all solution…

a. The zero vector is in H.... b. For each u and v in H, the sum…

The column space of a matrix A is the set Col A of all linear…

standard basis

What is the null space of matrix A? Wh…

The null space of a matrix A is the set Nul A of all solution…

A subspace of Rⁿ is any set H in Rⁿ th…

a. The zero vector is in H.... b. For each u and v in H, the sum…

What is a vector?

What is a matrix?

What is the vector space?

What are the 8 axioms of vector space?

Vector: a quantity with more than one element (more than one…

Put simply, a rectangular array of numbers, symbols, expressi…

A vector space (also called a linear space) is a collection o…

What is a vector?

Vector: a quantity with more than one element (more than one…

What is a matrix?

Put simply, a rectangular array of numbers, symbols, expressi…