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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 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.

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

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…

Three elementary row operations

Inconsistent

consistent

Every elementary row operation is reve…

1) replace one row by the sum of itself and a multiple of ano…

no solution

1) one solution only... 2) infinitely many solutions

True

Three elementary row operations

1) replace one row by the sum of itself and a multiple of ano…

Inconsistent

no solution

Linear equation

Linear system

linear system solution

Equivalent linear systems

an equation that can be written in the form a_1 x_1+a_2 X_2+.…

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

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

Linear systems are equivalent if they have the same solution…

Linear equation

an equation that can be written in the form a_1 x_1+a_2 X_2+.…

Linear system

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

A matrix is in reduced form if:

If a system has more variables than eq…

3 Facts about Linear Equations

Addition and Scalar Multiplication

1: Any rows consisting entirely of zeros are grouped together…

Must have an infinite number of solutions

Given an linear equation we must have 1) no solution... 2) one s…

Let u = (u1.....un) and v = (v1.....vn) be elements of Rn and…

A matrix is in reduced form if:

1: Any rows consisting entirely of zeros are grouped together…

If a system has more variables than eq…

Must have an infinite number of solutions

Linear Equation

System of Linear Equations

Solution

Solution Set

describes a firsrt order line in Rn equations of the form a1x…

collection of 1+ linear equations to be solved simultaneously

a set of real numbers s1...sn such that each equation holds i…

a set of all solutions in a system

Linear Equation

describes a firsrt order line in Rn equations of the form a1x…

System of Linear Equations

collection of 1+ linear equations to be solved simultaneously

equivalent

solution set

consistent

inconsistent

Two linear systems are called equivalent if they have the sam…

the set of all possible solutions of a linear system

A system of linear equations is said to be consistent if it h…

A system of linear equations is inconsistent if it has no sol…

equivalent

Two linear systems are called equivalent if they have the sam…

solution set

the set of all possible solutions of a linear system

A linear system is consistent iff...

If a linear system is consistent then…

Linear Combination

Span(v1...vp)

The rightmost column of the augmented matrix is not a pivot c…

A unique solution w/ no free variables or infinitely many sol…

Given vectors *v*1,*v*2,...**v**p **v**en vectors *v*1,*v*2,...**v**p in R^n and given scalars c1,c…

Is the collection of all vectors that can be written in the f…

A linear system is consistent iff...

The rightmost column of the augmented matrix is not a pivot c…

If a linear system is consistent then…

A unique solution w/ no free variables or infinitely many sol…

Definition of equality

Definition of a subspace

Definition of a transpose

Definition of a symmetric matrix

Two m ⨉ n matrices are equal if all of their corresponding en…

A subset W of a vector space V over a field F is called a sub…

The transpose of At of an m ⨉ n matrix A is the n ⨉ m matrix…

A symmetric matrix is a matrix A such that At = A.

Definition of equality

Two m ⨉ n matrices are equal if all of their corresponding en…

Definition of a subspace

A subset W of a vector space V over a field F is called a sub…

Linear Equation

System of Linear Equations

Solutions

Consistent

a1x1+a2x2+...+anxn=b

a11x1+a12x2+...+a1nxn=b1... a21x1+a22x2+...+a2nxn=b2... . ... .... .…

a sequence of n numbers that satisfies the equation.

A linear equation with a solution.

Linear Equation

a1x1+a2x2+...+anxn=b

System of Linear Equations

a11x1+a12x2+...+a1nxn=b1... a21x1+a22x2+...+a2nxn=b2... . ... .... .…

linearly independent

linearly dependent

if there is a free variable...

theorem 7

{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3…

{v1, v2, v3,... vn} is linear dependent if x1v1 + x2v2 + x3v3…

linearly dependent

if the set is linearly dependent at least one of the vectors…

linearly independent

{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3…

linearly dependent

{v1, v2, v3,... vn} is linear dependent if x1v1 + x2v2 + x3v3…

A linear transformation is a special t…

If A is an m x n matrix, then the rang…

If A is a 3x5 matric and T is a transf…

Every linear transformation is a matri…

True. The transformation is a function from one set of vector…

False. The range of T is the set of linear combinations of th…

False. The domain of T is Rn when A has n columns

False. Every matrix transformation is a linear transformation…

A linear transformation is a special t…

True. The transformation is a function from one set of vector…

If A is an m x n matrix, then the rang…

False. The range of T is the set of linear combinations of th…

span

An example of a linear combination of…

The solution set of the linear system…

The set span{u,v} is always visualized…

the collection of all possible linear combinations of v1, v2,…

True

True

False, if the vectors are linearly dependent then their span…

span

the collection of all possible linear combinations of v1, v2,…

An example of a linear combination of…

True

In a system of equations, each equatio…

In a system of equations, each variabl…

what are the 3 elementary transformati…

systems of equations are solved by

restriction

degree of freedom

1. add rows together... 2. switch order of rows... 3. multiply each…

systematically eliminating variables

In a system of equations, each equatio…

restriction

In a system of equations, each variabl…

degree of freedom

Elimination and Substitution

always

infinite

Linear equation

What are the two ways to solve for a linear system?

Homogeneous linear equations _________ have a solution

If two lines have the same slope, then the system has _____ a…

A ____ ____ of the variables x1,...,xn has the form... a1x1 + a…

Elimination and Substitution

What are the two ways to solve for a linear system?

always

Homogeneous linear equations _________ have a solution

how to tell if b can be written as a l…

span {v1...vp} is...

Matrix equation

A(u+v)=

combine into augmented matrix and see if the matrix is consis…

the collection of all vectors which can be written in the for…

[v1 v2 v3][x1 x2 x3] ... Ax=b

Au + Av

how to tell if b can be written as a l…

combine into augmented matrix and see if the matrix is consis…

span {v1...vp} is...

the collection of all vectors which can be written in the for…

Linear combination

Linear equation

System of linear equations

Solution

A _____ of A1, A2, ... An is an expression of the form :... c1A1…

A _____ is a linear cobination of unknowns set equal to a con…

A finite set of linear equations that all use the same variab…

A _____ to a system of linear equations is an assignment of v…

Linear combination

A _____ of A1, A2, ... An is an expression of the form :... c1A1…

Linear equation

A _____ is a linear cobination of unknowns set equal to a con…

THEOREM 7: The inverse of a matrix, if…

THEOREM 9: Let A and B be n × n invert…

THEOREM 10 If the n × n matrix A is in…

THEOREM 11 If A is an m × n matrix, th…

Proof: Assume that the square matrix A has an inverse and tha…

Proof Using the properties of matrix multiplication, we have…

In certain cases we can find the solution of a linear system…

If a homogeneous linear system Ax = 0 is such that A is inver…

THEOREM 7: The inverse of a matrix, if…

Proof: Assume that the square matrix A has an inverse and tha…

THEOREM 9: Let A and B be n × n invert…

Proof Using the properties of matrix multiplication, we have…

Theorem 1: Uniqueness of the Reduced E…

Theorem 2: Existence and Uniqueness Th…

Theorem 3: The Matrix Equation

Theorem 4: Existence of Solutions

Each matrix is row equivalent to one and only one reduced ech…

A linear system is consistent if and only if the rightmost co…

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

Let A be a m x n matrix. Then the following statements are lo…

Theorem 1: Uniqueness of the Reduced E…

Each matrix is row equivalent to one and only one reduced ech…

Theorem 2: Existence and Uniqueness Th…

A linear system is consistent if and only if the rightmost co…

linear equation

linear system

solution

types of solutions

variables x1, x2, x3... variables have to stand alone

collection of linear equations with same variables

list of numbers that make each equation true when substituted…

no solution (lines parallel), one solution (lines cross) or i…

linear equation

variables x1, x2, x3... variables have to stand alone

linear system

collection of linear equations with same variables

Theorem 1: Uniqueness of the Reduced E…

Theorem 2: Existence and Uniqueness Th…

Using Row Reduction to Solve a Linear…

Zero Vector

Each matrix is row equivalent to one and only one reduced ech…

A linear system is consistent if and only if the rightmost co…

Write the augmented matrix of the system

Vectors whose entries are zero

Theorem 1: Uniqueness of the Reduced E…

Each matrix is row equivalent to one and only one reduced ech…

Theorem 2: Existence and Uniqueness Th…

A linear system is consistent if and only if the rightmost co…

Row Echelon Form

Reduced Row Echelon Form

How many solutions?

leading variables

1. LC in every row is 1.... 2. LC in every row is to the right o…

SAME as REF except ... 4. All coefficients above LC's are zeroes.

- if zero= nonzero---> no solution... - # of LC= # of variables/…

variables that are represented by a column with a leading one

Row Echelon Form

1. LC in every row is 1.... 2. LC in every row is to the right o…

Reduced Row Echelon Form

SAME as REF except ... 4. All coefficients above LC's are zeroes.

linear equation

system of linear equations

solution

consistent system of linear equations

an equation that can be written in the form a₁x₁+a₂x₂+...

a collection of one or more linear equations using the same v…

a list of numbers (s₁,s₂...) that makes each equation a true…

a system of linear equations that has either one or infinitel…

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 using the same v…

N; Z; Q; R

x ∈ X

X ⊆ Y

X ⊂ Y

the sets of natural numbers, integers, rational numbers,... and…

denotes that x is a member of the set X

X is a subset of Y

X is a proper subset of Y

N; Z; Q; R

the sets of natural numbers, integers, rational numbers,... and…

x ∈ X

denotes that x is a member of the set X

m x n matrix

Is the system consistent?

Echelon Form

REF

m rows and n columns

Does at least one solution exist, if yes is it the only one,…

1. All nonzero rows are above any rows of all zeros... 2. Each l…

1. The leading entry in each nonzero row is 1... 2. Each leading…

m x n matrix

m rows and n columns

Is the system consistent?

Does at least one solution exist, if yes is it the only one,…

Non-Trivial Solution

Homogeneous equation

Describe x = p + tv

Trivial Solution

The homogenous equation Ax=0 has a nontrivial solution if and…

An equation in the form Ax=0, possibly written as a vector eq…

a line through p that is parallel to v

For a system of the form Ax=0 a zero solution

Non-Trivial Solution

The homogenous equation Ax=0 has a nontrivial solution if and…

Homogeneous equation

An equation in the form Ax=0, possibly written as a vector eq…

The equation Ax = b is referred to as…

A vector b is a linear combination of…

The equation Ax = b is consistent if t…

The first entry in the product Ax is t…

False. This equation is the general form of a matrix equation

True. This equation has a solution if and only if b is a line…

Ax = b is consistent if the coefficient matrix [A b] has a pi…

True

The equation Ax = b is referred to as…

False. This equation is the general form of a matrix equation

A vector b is a linear combination of…

True. This equation has a solution if and only if b is a line…

In some cases, a matrix may be row red…

The row reduction algorithm applies to…

A basic variable in a linear system is…

Finding a parametric description of th…

False. each matrix is row equivalent to one and only one redu…

False. The row reduction algorithm applies to any matrix whet…

True. The variables corresponding to pivot columns in an augm…

True.

In some cases, a matrix may be row red…

False. each matrix is row equivalent to one and only one redu…

The row reduction algorithm applies to…

False. The row reduction algorithm applies to any matrix whet…

linear equation

system of linear equations

solution to linear system

solution set

equation that can be written in the form: a1x1 + a2x2+... anx…

a collection of linear equations involving x's that you don't…

list (s1, s2, s3, ... sn) which satisfies each equation when…

set of all possible solutions

linear equation

equation that can be written in the form: a1x1 + a2x2+... anx…

system of linear equations

a collection of linear equations involving x's that you don't…

a __ is a collection of one or more li…

a __ for a SLE is a list of numbers su…

two systems are called __ if they have…

three of the elementary row operations

system of linear equations

solution

equivalent

replacement interchange scaling

a __ is a collection of one or more li…

system of linear equations

a __ for a SLE is a list of numbers su…

solution

If T: Rn maps to Rm is a linear transf…

A linear transformation T: Rn maps ont…

If T: R2 to R2 rotates vectors about t…

When two linear transformations are pe…

Then there exists a unique matrix A such that T(x) = Ax for a…

True. T is completely determined by what it does to the colum…

True

False. Linear Transformations preserve the operations of vect…

If T: Rn maps to Rm is a linear transf…

Then there exists a unique matrix A such that T(x) = Ax for a…

A linear transformation T: Rn maps ont…

True. T is completely determined by what it does to the colum…

the __ of a row refers to the leftmost…

a __ in a matrix A is a location in A…

a __ is a column of A that contains a…

pivot positions represent __

leading entry

pivot position

pivot column

basic variables

the __ of a row refers to the leftmost…

leading entry

a __ in a matrix A is a location in A…

pivot position

a matrix with only one column is calle…

two vectors in R^2 are __ if and only…

the vector whose entries are all zero…

given vectors v_1, v_2, ... , v_p in R…

column vector

equivalent

zero vector

linear combination weights

a matrix with only one column is calle…

column vector

two vectors in R^2 are __ if and only…

equivalent

C.The solution set is empty.

C. The solution set contains one solu…

A. The matrix is the augmented matrix…

Is the statement "Two matrices are ro…

A. The solution set has exactly one element, left parenthesi…

A. The solution has infinitely many elements.... B. The solutio…

Determine the value(s) of h such that the matrix is the augm…

C. False, because if two matrices are row equivalent it means…

C.The solution set is empty.

A. The solution set has exactly one element, left parenthesi…

C. The solution set contains one solu…

A. The solution has infinitely many elements.... B. The solutio…

Linear Equation

Linear System

Solution

Solution Set

An equation that can be written in the form:... a1x1+a2x2+...+an…

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

A list (s1, s2, ..., sn) of numbers that makes each equation…

The set of all possible solutions for a given system

Linear Equation

An equation that can be written in the form:... a1x1+a2x2+...+an…

Linear System

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

When is a system of linear equations c…

When is a system of linear equations i…

When do 2 linear systems have the same…

(Row) Echelon Form

When it has either one solution or infinitely many solutions

When it has no solution

If their augmented matrices are row equivalent

1) All nonzero rows are above any rows of all zeros... 2) Each l…

When is a system of linear equations c…

When it has either one solution or infinitely many solutions

When is a system of linear equations i…

When it has no solution

Consistent

Inconsistent

unique solution

row equivalent

a system of linear equations that has either 1 or infinte sol…

a system of linear equations that has no solution

the solutions when there is only one possible solutions for a…

when you can turn one matix into the other

Consistent

a system of linear equations that has either 1 or infinte sol…

Inconsistent

a system of linear equations that has no solution

conjecture

theorem or rule

equation

linear equation in one variable

an unproven statement about a general mathematical concept

a proven statement about a general mathematical concept

a statement that two expressions are equal

an equation that can be written in the form ax + b = 0, where…

conjecture

an unproven statement about a general mathematical concept

theorem or rule

a proven statement about a general mathematical concept

Vector

Physics definition

Computer Science definition

Mathematics definition

Is abstraction, the meaning of it depends on the particular a…

In this field of study, where vector represents something tha…

In this field of study, where vector represents a pair (or ev…

In this field of study, where vector is abstracted, so it rep…

Vector

Is abstraction, the meaning of it depends on the particular a…

Physics definition

In this field of study, where vector represents something tha…

linear equation

linear system

solution to a linear system with n var…

solution set

an equation of the variables x1, x2, . . . , xn that can be w…

a collection of one or more linear equations using the same v…

a list of n numbers (s1, s2, . . . sn)

the set of all possible solutions to a given linear system

linear equation

an equation of the variables x1, x2, . . . , xn that can be w…

linear system

a collection of one or more linear equations using the same v…

vector space

subspace

span

spans (generates)

a space V over a field F consists of a set which two operatio…

A subset W of a vector space V over a field F is a subspace o…

Let S be a nonempty subset of a vector space V. the span of S…

A subset S of a vector space V generates (or spans) V if span…

vector space

a space V over a field F consists of a set which two operatio…

subspace

A subset W of a vector space V over a field F is a subspace o…

Theorem 1.1 (Cancellation Law for Vect…

Theorem 1.2

Theorem 1.3

Theorem 1.4

If x,y, and z are vectors in a vector space V such that x+z=y…

In any vector space V, the following statements are true:... a)…

Let V be a vector space and W a subset of V. Then W is a subs…

Any intersection of subspaces of a vector space V is a subspa…

Theorem 1.1 (Cancellation Law for Vect…

If x,y, and z are vectors in a vector space V such that x+z=y…

Theorem 1.2

In any vector space V, the following statements are true:... a)…

What is a matrix?

What is a square matrix?

What is a diagonal matrix?

What is Equality

A matrix is a rectangular array of numbers or... symbols.

A matrix with the same number of columns and rows, also known…

A square matrix, N x N, that at entries that are not on the d…

Two matrices are equal if they are the same size and all the…

What is a matrix?

A matrix is a rectangular array of numbers or... symbols.

What is a square matrix?

A matrix with the same number of columns and rows, also known…