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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.

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.

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

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

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…

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.…

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…

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…

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…

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…

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…

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.)

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…

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

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…

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…

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.

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?

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

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 < >

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

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…

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…

Matrix Dimensions are given by? What d…

Scalar multiplication of a matrix?

How are Matrices added/subtracted?

What is the transpose of a matrix and…

rows x columns (m x n)... The rows are the number of equations…

A number that multiplies each number in the matrix.

Just add/subtract each corresponding entry.... They must theref…

You just swap rows with columns.

Matrix Dimensions are given by? What d…

rows x columns (m x n)... The rows are the number of equations…

Scalar multiplication of a matrix?

A number that multiplies each number in the matrix.

nothing, it is in standard form

9, 7, 4

10

3

What must to be done to put this expression in standard form?

What are the coefficients?

What is the constant?

What is the degree?

nothing, it is in standard form

What must to be done to put this expression in standard form?

9, 7, 4

What are the coefficients?

Order of Operations

Inequality

Like Terms

Equivalent expressions

PEMDAS the order in which you solve an equation.

An open sentence that contains the symbols < >

Terms that contain the same variables, with corresponding var…

Expressions that denote the same number.

Order of Operations

PEMDAS the order in which you solve an equation.

Inequality

An open sentence that contains the symbols < >

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…

Consistant

Coefficient matrix

Augmented matrix

Echelon matrix

A linear equation set with one or infinite solutions (imterse…

Only the coefficient of linear equations

Shows coefficients and adds a column with constants from righ…

1. All non zero rows are above all zero rows... 2. Each leading…

Consistant

A linear equation set with one or infinite solutions (imterse…

Coefficient matrix

Only the coefficient of linear equations

Linear Dependence

Linear Independence

Linear Combination

Span

When a vector in a set can be expressed as a linear combinati…

When none of the vectors in a set can be expressed as a linea…

c1v1 + c2v2 + c3v3 ... cnvn | ci is a member of R for i < n+1…

The set of all of the vectors that can be created by linear c…

Linear Dependence

When a vector in a set can be expressed as a linear combinati…

Linear Independence

When none of the vectors in a set can be expressed as a linea…

consistent system

inconsistent system

row operations

row echelon form

a system with at least one solution

a system with no solutions

multiply an equation by a non zero constant; add a multiple o…

first non zero entry in each row is 1 and every "leading 1" h…

consistent system

a system with at least one solution

inconsistent system

a system with no solutions

A linear equation in the variables x_1…

A solution of the system

The set of all possible solutions is c…

Two linear systems that have the same…

(a_1) ** X1 + (a_2) ** X2 + ... + (a_n) * Xn

is a list (s_1, s_2, ... , s_n) of numbers that makes each eq…

solution set

are called "equivalent"

A linear equation in the variables x_1…

(a_1) ** X1 + (a_2) ** X2 + ... + (a_n) * Xn

A solution of the system

is a list (s_1, s_2, ... , s_n) of numbers that makes each eq…