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invertible matrix

If A and B are invertible matrices of…

formula meaning A^n is invertible

If A is an invertible matrix, then

all A such that there exists A^-1 such that AA^-1 = A^-1A = I

AB is invertible, and (AB)^-1 = B^-1A^-1

(A^n)^-1 = (A^-1)^n

A^-1 is invertible, and A^n is invertible

invertible matrix

all A such that there exists A^-1 such that AA^-1 = A^-1A = I

If A and B are invertible matrices of…

AB is invertible, and (AB)^-1 = B^-1A^-1

linear equation

slope

y-intercept

x-intercept

An equation in two variables whose graph in a coordinate plan…

A ratio comparing the change in output over the change in inp…

The value of the output when the input is zero (0, y). The po…

The value of the input when the output is zero (x, 0). The po…

linear equation

An equation in two variables whose graph in a coordinate plan…

slope

A ratio comparing the change in output over the change in inp…

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.…

Variables... Section 1-1 variables and ex…

Algebraic expressions... Section 1-1 vari…

Factors... Section 1-1 variables and expr…

Product... Section 1-1 variables and expr…

Are symbols used to represent unspecific numbers or values. A…

Consists of one or more numbers and variables along with one…

The quantities being multiplied... Example 2x4

The outcome of what is being multiplied... Example 2x4 product i…

Variables... Section 1-1 variables and ex…

Are symbols used to represent unspecific numbers or values. A…

Algebraic expressions... Section 1-1 vari…

Consists of one or more numbers and variables along with one…

algebra

inconsistent system

matrix

square matrix

restoration (of broken parts)

a system with no solutions

rectangular array of numbers

a matrix in which the number of columns is equal to the numbe…

algebra

restoration (of broken parts)

inconsistent system

a system with no solutions

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…

In n variables; x1, x2, x3,....,xn has…

In in variables, is a set of m equatio…

1.1 DEF 1b: In this equation, a1, a2,.…

1.1 DEF 2a: A _______ of a linear equa…

linear equation

system of equations

-coefficients... -leading coefficients... -leading variable

-solution

In n variables; x1, x2, x3,....,xn has…

linear equation

In in variables, is a set of m equatio…

system of equations

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 System

Inconsistent System

Consistent Dependent System

Free variable

The matrix system has a solution

Matrix has no solution (last row 0 0 | non zero )

Infinite systems (last row 0 0 | 0 )

x1 x2 x3 etc can be any number, no set equation, no pivot in…

Consistent System

The matrix system has a solution

Inconsistent System

Matrix has no solution (last row 0 0 | non zero )

Linear equation

Linear system (system of linear equati…

Solutions

Solution set

Equation that can be written in the form a₁x₁+a₂x₂+... = b

Collection of one or more linear equations involving the same…

Numbers that make each equation true

Set of all possible solutions

Linear equation

Equation that can be written in the form a₁x₁+a₂x₂+... = b

Linear system (system of linear equati…

Collection of one or more linear equations involving the same…

System of Linear Equations

Solution of a System of Linear Equations

Solution Set of a System of Linear Equ…

Equivalent Systems

A collection of m equations with the same variable quantities…

The solution of a system is a list (s1, s2, s3, ..., sn) of n…

The set of all possible solutions for a linear system.

When two linear systems have the SAME solution set.

System of Linear Equations

A collection of m equations with the same variable quantities…

Solution of a System of Linear Equations

The solution of a system is a list (s1, s2, s3, ..., sn) of n…

(VS1) ... Commutativity of Addition

(VS2)... Associativity of Addition

(VS3)... Zero Vector;... Identity Element of…

(VS4)... Inverse Elements of Addition

For all x, y in V, x+y = y+x

For all x, y, z in V (x+y)+z = x+ (y+z)

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

For each element x in V there exists and element y in V such…

(VS1) ... Commutativity of Addition

For all x, y in V, x+y = y+x

(VS2)... Associativity of Addition

For all x, y, z in V (x+y)+z = x+ (y+z)

Thm 1

Thm 2

Linear Independence

Linear Dependence

A linear system has at least one solution if & only if the ri…

Let A be an mxn matrix coefficient matrix. Then the following…

The columns of A are linearly independent if & only if the tr…

The columns of A are linearly dependent if there is a nontriv…

Thm 1

A linear system has at least one solution if & only if the ri…

Thm 2

Let A be an mxn matrix coefficient matrix. Then the following…

Consistent System

Inconsistent System

Row Equivalent Matrices

Linear Combination

The system has either 1 solution or infinitely many

The system has no solutions

If a sequence of Elementary Row Operations can transform one…

Given vectors in R^n and scalars c1....cp, the vector b is gi…

Consistent System

The system has either 1 solution or infinitely many

Inconsistent System

The system has no solutions

Elementary row operations

Equivalent systems of linear equations

Matrix in row-echelon form properties

Matrix in reduced row-echelon form pro…

1. Interchange two rows... 2. Multiply a row by a nonzero consta…

Systems that have the same solution set

1. Rows of only zeros are at the bottom... 2. For each row that'…

It's in row-echelon form and every column that has a leading…

Elementary row operations

1. Interchange two rows... 2. Multiply a row by a nonzero consta…

Equivalent systems of linear equations

Systems that have the same solution set

Linear system

What are two basic methods of solving…

Two systems of linear equations are eq…

How do we know the method of eliminati…

A set of linear equations.

Method of elimination and substitution.

If they have the same solutions.

Each step in the method of elimination gives an equivalent sy…

Linear system

A set of linear equations.

What are two basic methods of solving…

Method of elimination and substitution.

(i,j)-entry of A

The diagonal entries of A

Main diagonal

Diagonal matrix

The entry in A in the ith row and the jth column.

a11, a22,a33...

The diagonal of A formed by the diagonal entries.

A square n x n matrix whose nondiagonal entries are zero.

(i,j)-entry of A

The entry in A in the ith row and the jth column.

The diagonal entries of A

a11, a22,a33...

Equivalent Systems

consistent system

inconsistent system

Row Echelon Form Qualifications [REF]…

Have the same solution set

either one solution or infinitely many

no solution

1. All zero rows below nonzero rows... 2. Each leading entry of…

Equivalent Systems

Have the same solution set

consistent system

either one solution or infinitely many

x bar parallel is

two vectors are orthogonal to each oth…

magnitude is

In an LU decomposition, the columns of…

the orthogonal projection of x bar onto the line

u * v = 0

the square root of each component squared

the pivotal columns in original matrix

x bar parallel is

the orthogonal projection of x bar onto the line

two vectors are orthogonal to each oth…

u * v = 0

rate or rate of change

constant rate of change

constant rate of change

zero rate

measure like miles per hour or price per item

rate that stays the same throughout the table

graph of this rate is a straight line

graph of this rate is a horizontal line

rate or rate of change

measure like miles per hour or price per item

constant rate of change

rate that stays the same throughout the table

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.

basic variable

existance

consistent

inconsistant

variable that corresponds to a pivot column in the coefficien…

there is at least one solution to the matrix

the system of equations has at least one solution

the system has no solutions

basic variable

variable that corresponds to a pivot column in the coefficien…

existance

there is at least one solution to the matrix

Linearly independent and linearly depe…

Linear dependence relation

The columns of matrix A are linearly i…

A set containing only one vector is li…

the equation A**x** = **0** has only the trivial solution.

that vector is not the zero vector.

Linearly independent and linearly depe…

Linear dependence relation

Linear Equation

Linear Coefficients

System of linear equations (linear sys…

Solution of a linear system

1. variables x1......xn that can be written in the form:... a1x1…

Can be real or complex numbers

Collection of one of more linear equations that contain x1...…

List (s1, s2.....,sn) of numbers that makes each equation a t…

Linear Equation

1. variables x1......xn that can be written in the form:... a1x1…

Linear Coefficients

Can be real or complex numbers

elimination method

linear inequality

solution of a system of linear inequal…

solution of a system of linear equations

solving systems by adding or subtracting equations to elimina…

an inequality in two variables whose graph is a region of the…

a set of values that make all inequalities in a system true.…

a set of values that make all equations in a system true. [p.…

elimination method

solving systems by adding or subtracting equations to elimina…

linear inequality

an inequality in two variables whose graph is a region of the…

The leading entry of the row is the:

Echelon Matrix Form

Reduced Echelon Matrix Form

Inconsistent/Consistent Matrices and E…

leftmost nonzero entry in that row

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

1. Same three reqs. as echelon form... 2. Leading entry is 1... 3.…

1. If there is a solution, there is a reduced echelon form to…

The leading entry of the row is the:

leftmost nonzero entry in that row

Echelon Matrix Form

1. All nonzero rows are above any rows of zeros... 2. "Stairstep…

Homogeneous Linear System

Trivial/Nontrivial Solutions

The homogeneous equation Ax=0 has a no…

A nontrivial solution x can have some…

1. A system of linear equations is said to be homogeneous if…

1. The zero solution is called the trivial solution... 2. The no…

the equation has at least one free variable.

not all of the entries are zero.

Homogeneous Linear System

1. A system of linear equations is said to be homogeneous if…

Trivial/Nontrivial Solutions

1. The zero solution is called the trivial solution... 2. The no…

Every linear transformation is...

Let T: R^n -> R^m be a linear transfor…

Standard matrix for the linear transfo…

Standard matrix for the linear transfo…

a matrix transformation (within finite dimensional spaces).

standard basis

Every linear transformation is...

a matrix transformation (within finite dimensional spaces).

Let T: R^n -> R^m be a linear transfor…

The number c in cu is a _________.

R^2 is a _______.

Vectors in R^3 are ______ X _______ co…

Vectors in R^3 are represented in ____…

scalar

rectangular coordinate system in a plane.

3; 1

three

The number c in cu is a _________.

scalar

R^2 is a _______.

rectangular coordinate system in a plane.

coefficient

distributive property

equation

inequality

the numerical part of a term

for any numbers a, b, and c: a(b+c) = ab+bc

a mathematical sentence showing two expressions are equal

mathematical sentences having the symbols <,>

coefficient

the numerical part of a term

distributive property

for any numbers a, b, and c: a(b+c) = ab+bc

Transformation/function/mapping /T

Domain

Codomain

Range

A rule that assigns to each vector **x** in R^n to a vector in…

The vector space R^n on which a transformation acts.

The vector space R^m that contains (but is no necessarily exh…

The set of all possible images T(**x**) of transformation T.

Transformation/function/mapping /T

A rule that assigns to each vector **x** in R^n to a vector in…

Domain

The vector space R^n on which a transformation acts.

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.…

A linear equation is...

A system of linear equations (a linear…

A solution to the linear system x_1, .…

Solving a linear system means...

variables x_1, x_2,...,x_n, any equation that can be written…

x_1,x_2,...,x_n a collection of linear equations that is the…

any ordered n-tuple (S_1,...,S_n) of real numbers if you subs…

to find the solution set, i.e. the set of all possible soluti…

A linear equation is...

variables x_1, x_2,...,x_n, any equation that can be written…

A system of linear equations (a linear…

x_1,x_2,...,x_n a collection of linear equations that is the…

Function

Function Notation

Input

Output

A relationship where each input has exactly one output.

If x is input and y is the output, then the function notation…

A number in the domain of a function. ... The independent variab…

A number in the range of a function.... The dependent variable.

Function

A relationship where each input has exactly one output.

Function Notation

If x is input and y is the output, then the function notation…

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 the augmented matrices of 2 linear…

Another way to ask if a system is cons…

Echelon Form (properties)

Reduced Echelon Form (properties)

the 2 have the same solution set

does at least 1 solution exist

All non zero rows above zero rows... each leading entry of a ro…

All properties from Echelon Form... leading entry is 1 in each…

If the augmented matrices of 2 linear…

the 2 have the same solution set

Another way to ask if a system is cons…

does at least 1 solution exist

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…

Consistent

Inconsistent

linear equation

system of linear equations

Either has one solution or infinitely many solutions

no solutions

in the variables x1...xn is an equation that can be written i…

collection of one or more linear equations involving the same…

Consistent

Either has one solution or infinitely many solutions

Inconsistent

no solutions