# Study sets matching "linear algebra chapter 1"

Study sets

Diagrams

Classes

Users

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

Absolute Value

Integer

Exponent

Function

a number's distance from zero

positive and negative whole numbers

tells how many times to multiply the base number

a relationship where the input determines the output

Absolute Value

a number's distance from zero

Integer

positive and negative whole numbers

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

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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 equ…

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 equ…

Linear Equation

System of Linear Equations

Solution of System

Linear Systems

a1x1+a2x2+...+anxn=b ... b and coefficients are real numbers

collection of one or more linear equations with the same varia…

Ordered list of numbers which make each of the equations true…

can be: ... -no sol'n (inconsistent)... -1 sol'n (consistent, unique…

Linear Equation

a1x1+a2x2+...+anxn=b ... b and coefficients are real numbers

System of Linear Equations

collection of one or more linear equations with the same varia…

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…

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 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

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

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…

C.The solution set is empty.

C. The solution set contains one solut…

A. The matrix is the augmented matrix o…

Is the statement "Two matrices are row…

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

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

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

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 parenthesis…

C. The solution set contains one solut…

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

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…

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…

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 (…

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…

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

linearly independent

linearly dependent

if there is a free variable...

theorem 7

{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3v…

{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 i…

linearly independent

{v1, v2, v3,... vn} is linear independent if x1v1 + x2v2 + x3v…

linearly dependent

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

In some cases, a matrix may be row redu…

The row reduction algorithm applies to…

A basic variable in a linear system is…

Finding a parametric description of the…

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

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

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

True.

In some cases, a matrix may be row redu…

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

The row reduction algorithm applies to…

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

A linear transformation is a special ty…

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

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

Every linear transformation is a matrix…

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

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

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 ty…

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

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

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

span

An example of a linear combination of v…

The solution set of the linear system w…

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 w…

span

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

An example of a linear combination of v…

True

The equation Ax = b is referred to as a…

A vector b is a linear combination of t…

The equation Ax = b is consistent if th…

The first entry in the product Ax is th…

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

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

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

True

The equation Ax = b is referred to as a…

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

A vector b is a linear combination of t…

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

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

A linear transformation T: Rn maps onto…

If T: R2 to R2 rotates vectors about th…

When two linear transformations are per…

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

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

True

False. Linear Transformations preserve the operations of vecto…

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

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

A linear transformation T: Rn maps onto…

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