Linear Algebra

Can you manipulate inequalities to make them into standard maximum form?

yes, you can multiply by -1 to flip the sign but you need to make sure that by doing so, the *right hand side* remains *positive* or else it is *NOT* in standard maximum form!

For maximization problems, if there are multiple non-basic variables that are negative which one do you choose?

you choose the row in which the basic variable is the MOST negative *we choose row 2 because -5000 is smaller is -3000*

How to know what side of the inequality to shade?

you shade *above* the line if the inequality is > you shade *below* the line if the inequality is < OR you pick (0,0) (0,1) (1,0) and if the coordinate makes the inequality *true* then shade *towards* the side *the coordinate is in*

When do you use a *dotted* line versus a *solid* line?

you use a *solid* when the inequality is greater than or equal to OR less than or equal to you use a *dotted* line when the inequality is just greater than or less than

TRUE vs FALSE Statement

0 = 0 is a *TRUE* statement 0 = 20 is a *FALSE* statement

how many cards are in each suit of cards?

13

What is a prime number?

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers ex: 2, 3, 5 etc

(A u B)' =

A' n B' *this also works with (A n B)'

What does it mean when you say that two sets are equal?

It means that they have the same elements ex: {1,4,5} = {5,4,1} **here duplicity and order doesn't matter, so {1,2,3} is still equal to {1,2,2,2,3,3}**

Why does the following system have infinite solutions? * x + y + z = 40 * *y + 3z = 49*

Signaled by the fact that there are *3 variables* but only *2 equations* all the variables depend on each other ex: make z = 3, then y = 43 and then x = -6

If you only compute interest *ONE* time what formula should you use?

Simple Interest Formula I = Prt

what suits are black in a deck of cards?

Spades and Clubs

Sinking Fund

a fund set up to receive periodic payments

boundary

a line that separates the point which are *in* the solution from the points which are *not in* the solution this is also the thing that splits a plane into 2 half planes

Column Matrix

a matrix with only one column

Row Matrix

a matrix with only one row ex: ( 1 2 3 )

inconsistent system

a system of equations that has no solution

Consistent System

a system of equations with *at least one* solution Ex: could be unique solution or infinite solutions

Simple Interest

a type of interest that is charged only on the amount borrowed

Annuities due

annuities in which payments are made at the beginning of each time period

Objective Function

function which we are asked to maximize/minimize

Constraints

restrictions of the function ie. x>9

Annuity

sequence of equal payments made at equal periods of time

When working with inequalities in max/min problems you never want to...

simplify the inequality ie. 3x + 3y < 6 SHOULD NEVER BE MADE x+ y <2

The sinking fund payment is the opposite of...

the future value of an ordinary annuity

A matrix that does not have an inverse is called

singular

When given info in Venn Diagram, you should always start with what info?

the information that is the intersection of everything (aka the *middlemost* section)

Identity Matrix

the matrix that has 1's on the main diagonal and 0's elsewhere *(A)(I) = (I)(A)* for it to be considered the identity matrix the identity matrix is considered to be like 1 --> you have XI=B(A^-1) this equals X = B (A^1) *(A)(inverse of A) = I*

If there are two coordinate pairs that each have the SAME Maximum or Minimum value then you say ....

the max or min is when x=5 and y =7 AND when x=3 and y=8 *AND all the points on the line segment in between*

When a matrix has no inverse what does this reveal?

the system has *no solutions* OR the system has *infinite number of solutions* To find out which one it is, you must do the Gauss Jordan Method (aka just solve the system normally)

Payment period

the time between payments

term

the time from the beginning of the first payment of the period to the end of the last period

Union of 2 Sets

the union of 2 sets is the set of all elements belonging to Set A and Set B. Denoted (A U B)

Shadow costs

they are the x values in the dual problem

What does it mean if 2 matrices are *equal*?

they have the same SIZE ---> same # of rows and columns they are corresponding elements ---> each element in each place is the same

A n (B u C) =

(A n B) u (A n C)

A' is a subset of what...

(A n B)'

A' u B' is equal to what

(A n B)'

For a CLOSED system what formula do you do?

(I-A)X = 0

For open systems, what formula do you do?

(I-A)^-1 x D

Subset

Set A is a subset of B if *every* element of A is also an element of B

what is the definition of an event?

an event is a subset of a sample space

What are natural numbers?

positive, whole #'s (NOT zero!!)

What is the form of a general solution?

(x, y) ex: if you have and equation y = 2x-1 then the general solution would be *(x, 2x-1)* **Always write the solution to a linear system in this form if there is more than ONE unique solution**

Present Value for compounding interest

**always check your answer to make sure it will produce the correct desired amount in x years as sometimes you may need to round up a cent**

Formula for finding the sinking fund payment

**this is the amount that you should pay periodically so you reach a certain goal** *AGAIN you may need to round up at times*

Echelon Method uses... Gauss-Jordan Method uses...

*Echelon Method:* variables (like keep in the x, y, z associated to the variables) *Gauss-Jordan Method:* augmented matrices

1) What is an upper Triangle Matrix? 2)What is a lower triangle matrix?

*Upper Triangle Matrix:* when all zeros are *below* the diagonal line *Lower Triangle Matrix:* when all zeros are *above* the diagonal line

What is a linear inequality by definition?

*a*x + *b*y < *c* where *a*, *b* can't *BOTH* be zero

How to read solutions from a dual problem

*only read off solutions if the element in the z column of the indicator row is 1* --> if it is not 1, then divide the indicator row to make it so You read off the entries of the slack variables in the indicator row and those entries will correspond with your y variables found in the original minimization problem ---> s1 will correspond with y1

you use *future value/sinking fund* when making monthly payments to ______________, while you use *present value/amortization payment* when trying to ___________

1) *Future value/sinking fund* corresponds to *SAVING $$* through periodic collections 2) *present value/amortization* corresponds to *PAYING BACK a loan* or *OWING $$*

What is the cardinality (# of elements) of each of the following: 1) ∅ 2) {∅}

1) 0 2) 1 since it is a set whose element is ∅

Simplex Method for Maximization Problem

1) Determine the objective function 2) Make the simplex tableau 3) Locate the left-most indicator --> if 2 indicators are equally both as negative, then choose the one farthest to the left 4) Form the necessary quotients, by dividing the RHS with the element in the same row of the column that houses the most negative element in indicator row. The *smallest* non-negative quotient gives the location of the pivot. --> if 2 indicators produce the same quotients which are equally as small, go with the row closest to the top --> disregard all quotients that have a *0* or a *negative #* in the *denominator* 5) Do row operations 6) If all the indicators are positive then you have found the final tableau. If not, repeat the method above until you have found the final tableau

In what cases should you check if you need to round up your results?

1) Finding the Present Value for Compound Interest 2) Sinking Fund Payments

How to solve a linear system using inverse matrices?

1) Make a matrix of coefficients (A) , of variables (B) and of solutions (C) the linear system shows (A)(B) = C so to isolate B (bc you want to find actual numbers for B as they now are only variable) *multiply A by its inverse* ---> (A)(A-1)(B) = (A-1)(C) --> B = (A-1) (C) 2) Find inverse of A and multiply it by the solutions matrix, C

7 steps of Echelon Method

1) Make sure that there is an x term in first equation, y term in second, etc 2) Eliminate *x* term in all equations after the *first* equation 3) Eliminate *y* term in all equations after the *second*equation 4) Eliminate *z* term in all equations after the *third* equation 5) Continue eliminating variables until there is a form *Ay= K* where A and K are constants ---> aka until it looks like 2y = 16 6) For each equation, make sure the first variable's coefficient is equal to 1 (if you want it in *reduced* echelon form) 7) Use substitution to find value of each variable

3 types of special matrices

1) Square Matrix 2) Row Matrix or Row Vector 3) Column Matrix or Column Vector

3 Ways of Transforming Systems

1) T1: exchange order of equations 2) T2: Multiply equation by a non-zero real number 3) T3: Add one equation to another --> if you have L1, L2, L3 (all of which are different equations) you can add a multiple of L1 to L2 and get an equation that is equivalent to L2)

Steps to solving a standard minimization problem

1) Write down the simplex tableau of the minimized problem BUT do NOT flip the sign of the indicators nor include the slack variable/Z columns! 2) take the transpose of the matrix stated above 3) Write the dual problem from the transpose --> when writing the objective function, just read directly from the indicator row... do NOT change the signs!! 4) find the max of the dual problem and this max will be the min of the original problem

How to find the Dual problem

1) Write the augmented matrix for the given problem 2) Take the transpose of the matrix 3) write the dual problem from that transpose

How to deal with Early Payments with loans?

1) You use the *Present Value for Ordinary Annuity* and make the number of payments (n) equal to the number of payments you have left. Ex: if you have 12 payments you need to make and choose to pay it off after 3 payments, then (n) would become *9* since 12-3 = 9.

Steps for solving a non-standard problem

1) convert the problem into a standard max problem (if necessary) 2) set z = -w if the problem was originally a *minimization* problem 3) add slack/surplus variables --> if inequality is < or equal to add surplus --> if equality > or equal to, add slack 4) write down simplex tableau 5) if any basic variable has a negative value, note the row of the non-zero entry 6) in the row located in step 5, find the column of the most positive entry that is farthest to the left 7) perform a pivot on that element 8) Continue until all basic variables are non-negative --> if this is impossible, then the problem has no feasible solution 9) Once a feasible solution has been found, the you can continue with the simplex method as usual to find the optimal solution

How to find the inverse of a 3x3 matrix?

1) find the *determinant* to see if the matrix is *invertible* 2) use the augmented matrix with the identity matrix on the left hand side and solve for the right hand side which will be the inverse

How to find the Vector Solution to a linear system

1) find the *general solution* to the linear system 2) make a column matrix for the variables, make a column matrix for the constant terms in each general solution, make a column matrix for the terms associated with the free variable but rename the free variable

What are 3 row operations?

1) interchange 2 rows 2) multiply elements of a row by any nonzero real number 3) adding a nonzero multiple of the elements of one row to the corresponding elements of a non-zero multiple of some other row

What the deal with Subsets and... 1) the empty set 2) the set itself

1) the empty set is always the subset of any set 2) the set is always a subset of itself`

Standard Maximum Form

1) the objective function is to be *maximized* 2) all variables are non-negative! --> aka there needs to be an initial statement saying that x1,> 0 etc 3) all remaining constraints are in the form *a*x1 + *b*x2 less than or equal to *c*

the 3 requirements for something to be in standard minimum form

1) the objective function is to be minimized 2) all variables are non-negative 3) all remaining constraints are in the form ay + by2 less than or equal to c when c is greater or equal to 0

How to find the range of optimal solutions or how to find ______

1) the slope of the objective function: F(x) = *a*x +*b*y --> *-a/b* 2) find the slope of each constraints: *a*x +*b*x = *c* --> *-a/b* 3) set the slope of the objective function in the middle of the slopes of the constraints --> let's say the slopes of the constraints are -2 and 7 then -2< -a/b < 7

How to Solve Linear Programming

1) write the *objective function* and *constraints* 2) *graph* the *feasible region* 3) Identify all *corner points* and evaluate the objective function @ each corner point --> the *max* will be the *highest* value of the objective function --> the *min* will be the *lowest* value of the objective function

How to Solve a Linear System using Augmented matrices?

1) write the linear system as an augmented matrix 2) use row operations to make sure all the elements in the 1st column except the one in the 1st row are 0s 2) use row operations to make sure all the elements in the 2nd column except the one in the 2nd row are 0s 3) continue this pattern until the last line is all 0s but the last column and the solution column ex: | 0 0 0 2 | 4| **if it asks for reduced echelon form then make sure all leading coefficients in each column are 1s**

Steps to solving a linear system in 2D

1. Multiply 1 or more equations until the same variable in both equations are eachother's least common multiple 2. Subtract one equation from the other until the other variable is isolated 3. Plug in the value of the isolated variable back into one of the original equation to find the value of the other variable

What are the 3 possible outcomes in a 2D linear system?

1. One *unique* solution (lines intersect) 2. No Solution (Parallel lines) --> *Inconsistent system* 3. Infinite Solutions (lines are the same) --> equations are *dependent*

If you divide the slack variable then it equals

1/(divided #) of the unused portion

Mutually Exclusive Events

2 events that can't occur at the same time --> when E n F is the empty set they are *disjointed sets*

how many cards are in a deck?

52

What symbol(s) do you use to show that something is/is not a member of a set?

ADD PICTURE

Reduced Echelon Form

Always have 1st coefficent of each row be 1 Preform matrix row operations to satisfy

What formula should be used? "The outstanding balance on Peter's credit card account is 3750 dollars. The bank issuing the credit card is charging 19.9 percent of interest per year compounded monthly. If Peter decides to pay off his balance in equal monthly installments at the end of each month for the next 11 months, how much will be his monthly payment?"

Amortization payment bc you OWE the credit card company money

The Empty Set

Can also be referred to as the *null set* Has no elements and is denoted with a zero with a line crossed through it

Compound interest is typically used with (longer/shorter) loans while simple interest is typically used with (longer/shorter) loans

Compound interest is used with *longer* loans Simple interest is used with *shorter* loans

How to find the determinant of a 3x3 matrix?

Don't forget to *ALTERNATE* the signs!!

How to find the determinant of a 4x4 matrix?

Don't forget to *ALTERNATE* the signs!!

General vs Particular Solution

General Solution: --> takes place in (x,y) form but with one variable replaced by an expression --> *ex: y= x+2 then (x, x+2) is general solution* Particular Solution: --> The general solution but with a substituted value --> *ex: y= x+2 and x=2 then the particular solution would be (2, 4)*

Transpose of a Matrix

Given an m x n matrix A, the transpose of A is the n x m matrix, denoted by A^T, whose cols are formed from the corresponding rows of A

what suits are red in a deck of cards?

Hearts and Diamonds

(A) x (A^-1) = ?

I (or the identity matrix)

Proper Subset

Set A is a proper subset of Set B if.. 1) Set A is a subset of B AND 2) Set A does not equal Set B

How to find the max of an unbounded region?

It doesn't have a maximum BUT the minimum will be one of the corner points

what are considered face cards?

Jack, Queen, King,

How to Multiply 2 Matrices together

Keep the row in the first matrix fixed and the multiply each of the elements in that row with each of the elements in the fixed column of the 2nd matrix

Nominal (or Stated) vs Effective Rate

Nominal interest is smaller than effective as the nominal interest rate is the amount of interest you would have to pay in a year if the interest rate was only compounded *once*

we can only add/subtract 2 matrices if they are the same ______

SIZE ---> same # of rows and columns

How to verify if 2 matrices are inverses of one another?

Step 1: Multiply the matrices together ---> if you have matrix A & matrix B do BA and AB Step 2: if *both* products are the identity matrix then they are inverses

Equivalent System

Systems that have the EXACT same solution set *Always check that the solutions to each system work for BOTH systems or else they are not equivalent* ex: if one equation's solution is (1,2) and that solution makes the 2nd equation true BUT the second equation has infinite solutions then they two systems are NOT equivalent bc equation 1 only has one unique solution rather than an infinite amount

TRUE OR FALSE: the original equations in a system can be solved by both particular and general solutions?

TRUE

TRUE or FALSE: An experiment can have more than one sample space?

TRUE, but try to find the sample space where all of the outcomes are equally likely

The Multiplicative Inverse of a Matrix

The multiplicative inverse of square matrix A, if it exists, is notated A-1, where the product of A and A-1 is the identity matrix. (A) x (inverse) = Identity Matrix

set-builder notation

This notation is used to show a common property of all elements in a set {x| x has property P} ex: for {x| x is a natural number less than 5} --> S = {1,2,3,4}

How to find the time for price doubling?

Use the compound interest formula but set *A=2* and *P=1*

The Gauss-Jordan Method

Using row operations to convert a matrix into reduced row echelon form

How to calculate how much you are paying towards interest in regards to equal monthly payments?

You find how much you *pay each month* and then *multiply* that by the *number of payments to be made*. THEN *subtract that* from the *amount of the loan* you took out

When you have the interest rate at 0%, how do you find the monthly payments?

You just divide the total monetary amount by the number of months

How to find the number of *non-negative, integer* solutions (aka the number of workers) if there are infinite amount of solutions?

You make sure that all the variables that are NOT the dependent are set > or equal to 0 and then solve. if general solution is (2y - 9, y) then you say that WHAT? 2y -9 > 0 which simplifies to y > 9/2 so the maximum amount of workers is 4 as 4.5 works is impossible and 5 workers does not comply with the system

When to use *Sinking Fund* Payments vs when to us *Amortization Payments*

You use *sinking fund payments* when you want to pay monthly payment so that you can *reach a certain goal*.. ie, your payments are working to collectively accumulate enough money so that you can have a certain desired amount of money in x amount of years. In this case, interest is working for you --->Ex: How much money would Jenny have to put into her savings account at the end of every 3 months to accumulate $150,000 over 79 years if she received an interest rate of 5.25% annually You use *amortization payments* when you need to *pay off a loan* and you want to make those payments in equal amounts. In this case, interest is working against you ---> Ex: Find the amount of monthly payment needed to amortize a loan of $220,000 for 30 yrs at an annual interest rate of 6%

What formula would you use to solve this: "Jane made a down payment of 3000 dollars toward the purchase of a car. To *pay the balance* of the purchase price, she has secured a *loan* from her bank at the nominal *rate of 4.5 percent per year compounded monthly*. Under the terms of her finance agreement, she is required to to make *payments of 270 dollars per month for 30 months*. What is the cash price of the car?

You would use the *present value of an ordinary annuity formula* bc you know you took out a lump sum (that will not change) so you want to find what that lump sum is... if you used the tempting *future value of an ordinary annuity* it would be incorrect bc then you would be gaining interest rather than paying interest "The present value of an annuity represents the sum that must be invested now to guarantee a desired payment in the future, while the future value of an annuity is the amount to which current investments will grow over time." ---> future value estimates how much you will save overtime *you use future value/sinking fund when making monthly payments to SAVE x amount of money, while you use present value/amortization payment when trying to PAY back a loan or OWE $$*

what is the definition of a *set?*

a well defined collection of objects in which it is possible to determine if a given object is included in the collective You use squiggly brackets to make the boundaries of a set

How to tell if a simplex tableau is in final form?

all the indicators (aka the last row of the simplex tableau) are either *positive* or *zero*

Square Matrix

an *n* x *n* matrix where the number of rows *equals* the number of columns

what is the definition of an experiment?

an activity or occurrence with an observable result

In a word problem situation when you are asked to multiply 2 matrices together, why is generally one of the options useless? ex: you have matrix F & matrix P why is either FP or PF useless?

because if matrix F is a 2 x 3 matrix that represents (location/shoe type) and matrix P is a 3 x 2 matrix that represents (shoe type/ owner) FP would be the *useful* matrix as the resulting 2x2 matrix would be (location/owner) PF would be the *useless* matrix as the resulting 3x3 matrix would be (shoe type/shoe type)

In a financial math question if you see the word indefinite, what type of compounding should you think of?

continuous compounding

what is the definition of a trial?

each repetition of an experiment

the intersection of a set with the empty set is the...

empty set

If you ever want to find the smallest or largest quantity of some thing...

ex: if you have forks, knives, spoon and x= fork, y = knife, z= spoon. If you want to find the smallest/largest quantity of *Spoons* then make sure z becomes the parameter (aka you make it last in the equation) aka you set it up in terms of the specific varible (x in terms of z, y in terms of z, z_)

Link for More Practice Problems

http://www.math-exercises.com/equations-and-inequalities/systems-of-linear-equations-and-inequalities

How to tell if a simplex tableau has no solution

if a column is *all negative* then there is no maximum solution!

How to tell if the outcomes are equally likely?

if all things have the same chance of occurring ex: the sample space {3 boys, 2 boys 1 girl} had outcomes that are NOT all equally likely bc there is only 1 chance of getting 3 boys (bbb) while there are 3 chances of getting 2 boys & 1 girl (bbg, gbb, bgb)

what is the definition of a certain event?

if an event E equals the sample space --> when the probability is 1 (aka 100%) ex: in a class that has 30 females and 0 males, the probability of choosing a female student

Corner Point Theorem

if an optimum value (a max/min) of the objective function exists, it will occur at one or more corner points of the feasible region

Ordinary Annuity

if payments are made at the *end* of the timer period and the frequency of payments equals the frequency of compounding

How to check if a matrix is invertible?

if the determinant equals *ZERO* then no inverse exists! *aka the matrix is NOT invertible*

How to tell if a simplex tableau needs further pivoting?

if there are any negative numbers in the indicator row, then the simplex tableau needs further pivoting

Example on how to make the solution logical

if x and y represent the # of female/male workers then they can't be negative numbers

Basic Probability Principle

in a sample space, S, of equally likely outcomes and event E is a subset of S, then the probability of E is... p(E) = n(E)/n(S)

How is a solution of an equation written?

in parenthesis *with only 2 unknowns ex: x= and y =2 in correct form would be (1,2)* *with more unknowns ex: x1 =2, x2=3, x3=7 in correct form (2,3,7)

Ordinary interest

interest found using a 360-day year

Exact interest

interest found using a 365-day year

Complement of a Set A

it is everything in the universal set BUT Set A denoted with A'

The initial Simplex Tableau shows our solution at what?

it shows our solution at the *origin* because the slack variables are the *basic variables* so all *other variables* (ie the x variables) will be *zero*, thus we find ourselves at the *origin*

first degree equations are also called

linear systems

Outstanding loan amount

loan amount which has to be repaid by the borrower to the bank on a particular date *not really needed*

what does an *m* x *n* matrix mean?

m = # of rows n = # of columns

What notation do we use to help keep track of the quantities a matrix represents?

meaning of the rows / meaning of the column *for example in picture the notion would be "models/styles"*

Scalar Multiples of a Matrix --> (aka multiply a matrix by a constant, or number)

multiply each entry in the matrix by the scalar

What does "n" equal in financial math questions?

n equals the number of payments to be made ex: if you compound something monthly and the time frame given in 11 months, then n = 11 --> if you compound something monthly and the time frame given is 5 years, then n = 60 since for each year you make 12 payments so over 5 years you make 12 x 5 = 60 payments

What is the Union Rule for Sets

n(A u B) = n(A) + n(B) - n(A n B)

(A U B U C) =

n(A) + n(B) + n(C) - n(A n B) - n(A n C) - n(C n B) + n(A u B u C)

Does a matrix with a row of all 0s have an inverse?

no, a row of all 0s makes it impossible to get all the 1s in the diagonal of the identity matrix this is bc of the fact that *having a row of all zeros means that the determinant would be 0*

Slack Variables

non-negative variables added to linear inequalities to make it into a linear equality. They are denoted by the variable "s". The *number* of *slack variables* EQUALS the *number* of *constraints*

When doing augmented matrices where should you indicate what row operation you preformed?

on the left hand side next to the row that is being manipulated

Only what type of matrix can have an inverse?

square matrices

if A is a subset of B, then what is true about their complements?

that B' is a subset of A'

When you multiply one matrix that is invertible by an unknown matrix and the solution is a matrix full of zeros, then what do you know about the unknown matrix?

that it ALSO can't be invertible

What does "the system is m equations in n unknowns" means?

that n = # of variables and m = # of equations

When forming quotients, if all quotients must be disregarded then what does this reveal?

that there is no maximum solution

Basic variables

the "m" amount of *non-zero* variables. The *amount of basic variables* EQUALS the *amount of equations* These are the variables that you solve for once you set all the other variables to zero In a simplex tableau, they are found when all entries but ONE in a column are zeros

what is the determinant of the transpose matrix equal to

the determinant of the original matrix so you just flip the rows and columns and then take the determinant as you normally would

Set Difference

the elements of one set excluding the elements of another set. Denoted (A\B)

what are the elements of the set S = {1,2,3}?

the elements/members of the set are 1, 2, 3

Pivot entry

the entry that you choose in a simplex tableau and then you do *row operations* to make *all entries* in that *column* BUT that particular entry *zero*

how to find the number of subsets there are for a particular set?`

the number of subsets is 2^n where n = # of elements of the set

Theorem of Duality

the objective function of a problem in standard minimum form (denoted w) takes on a minimum value if and only if the dual problem takes on a max value for Z

what is the definition of an outcome?

the possible results of each trial

the amortization payment is the opposite of...

the present value of an ordinary annuity

What is the solution of the system of inequalities

the region made up of all points which satisfy all the inequalities at the same time this is known as the *feasible region*

Non-basic variable

the remaining variables that are NOT the basic variables. They are identified in the simplex tableau bc their column is filled with random non-zero numbers (but they can also have zeros!) You automatically set these variables equal to zero *The amount of non-basic zero EQUALS the number of variables*

In general which variable is the parameter?

the rightmost one

what is the definition of a sample space?

the set of all possible outcomes for an experiment ex: the sample space of tossing a coin is {h,t}

Universal set

the set that includes all objects being discussed ex: when talking about rolling a die, the universal set is {1,2,3,4,5,6}

Intersection of Two Sets

think *n*tersection --> A n B the intersection of the two sets are all the elements that belong to both Set A and Set B --> ie. it's made up the elements in which Set A overlaps with Set B

With matrices, row operations are similar to __________ in linear systems?

transformations

Disjoint sets

two sets that have no elements in common --> their intersection is the empty set

Find the probability of rolling an even number on a die

using the basic probability principle.. p(E) = 3/6 since the number of even numbers are {2,4,6} and the universal set is {1,2,3,4,5,6}

From the soultions of a dual problem, how do we check if we have found a corner point?

we have found a corner point if the coordinate solution makes at least 2 of our inequalities (from the original into equalities

what does E u F mean

when E *or* F or both occur

what does E' mean

when E does NOT occur

What is an unbounded feasible region?

when a region is not fully enclosed

what is the definition of a simple event?

when an event has only one possible outcome

What does E n F mean

when both E and F occur

Compound interest

when interest is charged on both the interest and the principle amount

what is the definition of an impossible event?

when the Event = the empty set --> when the probability of an event is 0

continous compounding

when the number of times the interest is compounded becomes infinite

What is a bounded feasible region?

when the region is enclosed by lines of all sides these lines can be either *dotted* or *solid*

Integer Programming

when the solution to a linear programming problem is restricted to integers aka all fractional solutions are meaningless You have this when the solution you are looking for must be in whole numbers like # of workers bc you can't logically have 0.5 of a human!

x1, x2, x3 is another way to say

x, y, z *Aka they stand for variables*

The equation x = y+ 3 has a general solution of (y+3, y). Which variable is the parameter?

y