# 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 ro…

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 have…

Solution set

Linear systems can have ___________ sol…

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 rows…

If two systems of linear equations have…

they are equivalent

Solution set

all possible solutions to linear equation

Vector

Problem with definitions (6th Grade Exa…

Infusing Tech into Math Article

Velocity

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

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

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

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

Vector

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

Problem with definitions (6th Grade Exa…

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

affine transformation

augmented matrix

back substitution

basic variable

similar to a linear transformation, but does not necessarily p…

a matrix made up of a coefficient matrix for a linear system a…

phase of row reduction which transforms an echelon matrix to a…

a variable in the linear system that corresponds to a pivot co…

affine transformation

similar to a linear transformation, but does not necessarily p…

augmented matrix

a matrix made up of a coefficient matrix for a linear system a…

T

F

F

T

Every elementary row operation is reversible.

A 5x6 matrix has 6 rows.

The solution set of a linear system involving variables x1, ..…

Two fundamental questions about a linear system involve existe…

T

Every elementary row operation is reversible.

F

A 5x6 matrix has 6 rows.

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 collect…

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

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

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 collect…

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 _____ am…

A ____ ____ of the variables x1,...,xn has the form... a1x1 + a2…

Elimination and Substitution

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

always

Homogeneous linear equations _________ have a solution

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 wh…

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 wh…

(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 o…

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 th…

(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 o…

Three elementary row operations

Inconsistent

consistent

Every elementary row operation is rever…

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

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 anot…

Inconsistent

no solution

Theorem 1: Uniqueness of the Reduced Ec…

Theorem 2: Existence and Uniqueness The…

Using Row Reduction to Solve a Linear S…

Zero Vector

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

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

Write the augmented matrix of the system

Vectors whose entries are zero

Theorem 1: Uniqueness of the Reduced Ec…

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

Theorem 2: Existence and Uniqueness The…

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

equivalent

solution set

consistent

inconsistent

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

the set of all possible solutions of a linear system

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

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

equivalent

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

solution set

the set of all possible solutions of a linear system

Linear Equation

System of Linear Equations

Solution

Solution Set

describes a firsrt order line in Rn equations of the form a1x1…

collection of 1+ linear equations to be solved simultaneously

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

a set of all solutions in a system

Linear Equation

describes a firsrt order line in Rn equations of the form a1x1…

System of Linear Equations

collection of 1+ linear equations to be solved simultaneously

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

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

Matrix equation

A(u+v)=

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

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

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

Au + Av

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

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

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

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

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

1.3.15 When u and v are nonzero vectors…

1.3.15 Any list of five real numbers is…

1.3.15 Asking whether the linear system…

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

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

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

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

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

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

1.3.15 When u and v are nonzero vectors…

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

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 cons…

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

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

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 cons…

THEOREM 7: The inverse of a matrix, if…

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

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

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

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

Proof Using the properties of matrix multiplication, we have (…

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

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

THEOREM 7: The inverse of a matrix, if…

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

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

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 in…

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 Ec…

Theorem 2: Existence and Uniqueness The…

Theorem 3: The Matrix Equation

Theorem 4: Existence of Solutions

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

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

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 log…

Theorem 1: Uniqueness of the Reduced Ec…

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

Theorem 2: Existence and Uniqueness The…

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

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, i…

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

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, i…

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 of…

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

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

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 of…

Reduced Row Echelon Form

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