#### Study sets matching "linear algebra chapter 1"

#### Study sets matching "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

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 two systems of linear equations hav…

Solution set

Linear systems can have ___________ so…

Elementary Row Operations

they are equivalent

all possible solutions to linear equation

no, exactly one, infinitely many

add a multiple of one row to an existing row, interchange row…

If two systems of linear equations hav…

they are equivalent

Solution set

all possible solutions to linear equation

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…

What is a linear equation?

What is a system of linear equations?

What is a solution of a linear equation?

What is a solution set?

Any equation that can be written in the following form: (pg. 2)

A system of linear equations (or a linear system) is a collec…

A list of numbers that satisfies a system of linear equations…

The set of all possible solutions is called the solution set…

What is a linear equation?

Any equation that can be written in the following form: (pg. 2)

What is a system of linear equations?

A system of linear equations (or a linear system) is a collec…

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

(vs 1)

(vs 2)

(vs 3)

(vs 4)

for all x,y in V, x + y = y + x (commutativity of addition)

for all x,y,z in V, (x + y) + z = x + (y + z) (associativity…

there exists an element in V denoted by 0 such that x + 0 = x…

for each element x in V there exists an element y in V such t…

(vs 1)

for all x,y in V, x + y = y + x (commutativity of addition)

(vs 2)

for all x,y,z in V, (x + y) + z = x + (y + z) (associativity…

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…

Solution Set

Solution

Consistent

Inconsistent

Set containing all possible solutions of a linear system

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

If a system has either one or infinitely many solutions

If a system has no solution

Solution Set

Set containing all possible solutions of a linear system

Solution

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

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…

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…

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

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…

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…

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…

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…

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

1.3.15 The set Span {u,v} is always vi…

1.3.15 When u and v are nonzero vector…

1.3.15 Any list of five real numbers i…

1.3.15 Asking whether the linear syste…

False, if the vectors are LD then we visualize as a line not…

False, Span {u,v} contains a whole plane through the origin,…

True, the R stands for the real numbers that appear as entrie…

True, if b Spans {a1 a2 a3} then the linear system is consist…

1.3.15 The set Span {u,v} is always vi…

False, if the vectors are LD then we visualize as a line not…

1.3.15 When u and v are nonzero vector…

False, Span {u,v} contains a whole plane through the origin,…

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

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.

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,…