SUMMER COURSE OUTLINE IN LINEAR ALGEBRA

1. SETS AND ITS OPERATION

2. VENN-EULER DIAGRAM

3. APPLICATION OF SET

4. RELATIONS AND FUNCTIONS

5. COMPOSITION OF FUNCTIONS

6. REVIEW OF LONG METHOD DIVISION OF POLYNOMIALS 7. POLYNOMIAL THEOREMS

8. EXPONENTIAL EQUATIONS

9. LOGARITHMIC EQUATION

10. SOLVING LINEAR EQUATIONS IN TWO VARIABLES UNKNOWN

USING ELIMINATION AND SUBSTITUTION

11.WORDED PROBLEMS A. INTEGER PROBLEM

B. AGE PROBLEM

C. MOTION PROBLEM

D. CLOCK PROBLEM

E. MIXTURE PROBLEM

F. MONEY PROBLEM

G. INVESTMENT PROBLEM H. WORK PROBLEM

12.EXPONENTIAL GROWTH AND DECAY

13.MATRICES

A. GAUSSIAN ELIMINATION METHOD

B. DETERMINANTS AND CRAMERS‟ RULE

Module 1 : SETS and OPERATION

Set- a well-defined collection of objects, concrete or abstract of any kind.

TWO METHODS OF WRITING A SET

1. ROSTER METHOD

2. SET- BUILDER NOTATION

2 Major Type

1. Finite set- a set whose elements are limited or countable and the last element is

identifiable.

2. Infinite Set- a set whose elements are unlimited.

OTHER TYPE OF SET

1. EMPTY SET/ NULL SET or { }

2. UNIT SET {1}

3. UNIVERSAL SET U= {0,1,2,3,4,5,…20.}

4. SUBSET A= { 1,2,3,4,5,6,7,8,9,10.} B= { 2,3,4,6,8.} B A

5. EQUAL SET A= { L,I,S,T,E,N.} B= { S,I,L,E,N,T} B A

6. EQUIVALENT SET A= { 1,2,3,4} B= { A,B,C,D}

7. JOINT SET A={ 1,2,3,4,5,6,7,8,9,10.} B= { 2,3,4,6,8}

8. DISJOINT SET A={ 1,2,3,4,5,6,7,8,9,10.} B = { 0,11, 12,13,14…}

BASIC NOTATION

1. - INTERSECTION

2. - UNION

3. - NOT AN ELEMENT OF

4. - AN ELEMENT OF

5. - PROPER SUBSET OF

6. - IS EQUIVALENT TO

7. NULL SET

8. „ PRIME SYMBOL

NAME: _______________________________________ SCORE_________

DATE: _______________

EXERCISE 1.1: SET AND BASIC NOTATION

DIRECTIONS: IDENTIFY EACH STATEMENT AS TRUE OR FALSE

GIVEN A={ 2,5,8,11} and B={ 8,11,14,2}

1.

2.

3.

4.

5.

6. * +

7. * +

8. * +

9. {14,2}

10.* +

MODULE 2: OPERATIONS ON SET

THERE ARE 5 OPERATIONS ON SET SYMBOL USED

1. UNION OF SET

2. INTERSECTION OF SET

3. CARTESIAN PRODUCT AXB

4. COMPLEMENT OF A SET A‟( PRIME)

5. DIFFERENCE OF A SET A-B

UNION OF SET

Combined elements found in a given set with the other set.

A= { 1,2,3,4,5,6,7,8,9,10} B= { 2,5,7,9} * +

ITERSECTION OF SET

Element(s) which that are common to both given sets.

A= { 1,2,3,4,5,6,7,8,9,10} B= { 2,5,7,9} * +

CARTESIAN PRODUCT

Each Element of a given set A are being paired with each elements found in Set B.

A= { 1,2,3} B= { 2,5,7,9}

AXB= { (1,2), (1,5) ,(1,7), (1,9),( 2,2).(2,5),(2,7),(2,9), (3,2),(3,5),(3,7),(3,9)}

COMPLEMENT OF A SET

Elements which are found in the universal set but not in the given set.

U= { 0,1,2,3,4,5,6,7,8,9,10…15} A‟ = { 2,4,5,7,8,10,11,12,13,14,15}

A= { 0,1,3,6,9}

NAME: _________________________________________SCORE_____________

DATE: _________________

Exercise 1.2 OPERATIONS ON SET

DIRECTIONS: Given the sets below, give the elements of the following operations on set.

U= {1,2,3,4,5,6,7} A= {1,3,5,7} B= { 2,4,6} C= {1,3,5}

1. = {2,4,6} or B

2. = A

3. ‟= {2,4,6,7}

4. U

5. B

6. { }

7. ( ) U

8. ( ) { }

9. { }

10. C‟

NAME: _________________________________________

SCORE_____________

DATE: _________________

Exercises 1.3A VISUAL REPRESENTATION OF OPERATIONS ON SET

TAKE HOME QUIZ

DIRECTIONS: Given the sets below, give the elements of the following operations on set

using

Venn- Euler diagram and Shade the corresponding region where the elements are found.

Given; U= {1,2,3,4,5,6,7} A= {1,3,5,7} B= { 2,4,6} C= {1,3,5}

1.

2. =

3. * + =

4.

5.

6. ( )

7. ( )

8.

9.

10.( )

EXAMPLE

U= { 1,2,3,4,5,…20}

A= {1,2,3,4,5,6}

B= { 4,6,9}

4

6 9

1

2

3

7 8 20

A B

10

11 5 19

12 13 14 15 16 17 18

U

NAME: _________________________________________ SCORE_____________

DATE: _________________

EXERCISES 1.3B APPLICATION OF SETS

VENN- EULER DIAGRAM

DIRECTIONS: Read and analyze the following problem s below and represent the following set to solve

using the Venn- Euler diagram. Show your solution. (30 points)

1. A survey on subjects being taken by 250 college students in Metro Manila revealed

the following information; 90 likes Math; 88 English; 145 Filipino; 25 Math &

Filipino; 38 Filipino & English; 59 Math and English; 15 All. How many did not

take the survey?

2. There are 20 seniors serving the student council of CDNSHS. Of these, 3 have not

served before. 10 served on the council in their junior years, 9 in their sophomore

years, and 11 in their freshmen years. There are 5 who served during both their

sophomore and junior years, 6 during both freshman & junior years and 4 during

both freshman and sophomore years. How many seniors served in the student

council during each of the 4 years in high school?

3. In a survey involving 800 employees, it was found that 485 employees saves in

BDO , 550 save in Metro Bank , 540 save in BPI, 255 save in BDO and Metro

Bank. 270 save in BDO and BPI, and 325 save in Metro bank and BPI. All

employees save with at least one of these banks.

a. How many employees save in BPI but not in Metro Bank?

b. How many employees save in BDO but not in Metro Bank or BPI?

c. How many employees save in all three banks?

MODULE 3 RELATIONS AND FUNCTIONS

Definition of terms

A relation is any set of ordered pairs ( x,y), ( domain, range)

Relation = {x/x is any ( x,y)} i.e., { (1,2), (2,3), ( 4,5), ( 6,7)}

A relation S is a function if and only if (a,b) S and ( a,c) S implies that b=c

A function is a set of ordered pairs ( x,y) such that for each first component, there is

at most one value of the second component.

A function is a set of ordered pairs having the property that no two distinct

ordered pairs have the same first entry or domain.

NOTE:

All functions are relations but not all relations are functions.

A graph represents a function if and only if no vertical line intersects the graph

more than once.

y

0

1

2

3

3

3

y

1

2

3

4

5

NAME: _________________________________________

SCORE_____________

DATE: _________________

EXERCISES 1.4 RELATIONS AND FUNCTIONS

A. Directions: Determine whether the following relations are function or not. Write

F if it is a function and N if it is not a function.

______1. { ( 2,3), ( 4,4), ( 2,4), ( 3,2)}

______ 2. { ( 1,2), (2,3), (-1,2),( -2,3)}

______3. {0.2, 0.003), (0.002, .005), (2/10, 1/3)}

______4. { (

), (

, ), (

) }

_____5. { (1.5, -1.5), ( 2.5, -2.5), ( 3,3), ( -2,3)}

B. Directions: Determine the domain and range. Use the vertical line test to

determine whether the relation is a function or not.

Domain Range

1. _______ __________

2. _______ __________

3. _______ __________

4. _______ __________

5. _______ __________

C. Directions: In each of the following, indicate the values of x must be excluded from

the domain.

1. ( )

( )

2. ( )

( )

3. √

4. √

5. ( )

OPERATIONS ON FUNCTIONS (MDAS)

Let f and g be two functions with domains Df and Dg respectively. Then

1. ( )( ) ( ) ( )

2. ( )( ) ( ) ( )

3. ( )( ) ( ) ( )

4. (

) ( )

( )

( )

COMPOSITE FUNCTIONS

Let f and g be functions. The composition of f on g is the function defined by ( )( )

( ( )) where its domain is dom ( ) *

( ) ( ) ( ) +

( ) ( ( ))

( ) ( ( ))

OPERATIONS ON FONCTIONS LET f(x) = g(x) =

1. ( )( ) ( ) ( )

=

=

=

2. ( )( ) ( ) ( )

= ( )

=

3. ( )( ) ( ) ( )

= ( )( ) =

4. (

) ( )

( )

( )

=

( )( )

=

MODULE 4: POLYNOMIAL THEOREMS

REMAINDER THEOREM

If a polynomial f(x) is divided by (x-r) until a remainder independent of x is obtained then, the remainder is equal to P(r).

FACTOR THEOREM

If (x-r) is a factor of P(x), then r is a root of the equation P(x) =0.

FUNDAMENTAL THEOREM OF ALGEBRA If P(x) is a polynomial with positive degree, then P(x) has at least one zero.

THE NUMBER OF ROOTS THEOREM

IF P(x) is a polynomial of degree n, then P(x) =0 will have n roots.

Descarte‟s rule of sign

This states that for every change in the sign in a given polynomial this corresponds to the number of positive and negative roots.

Example

FUNDAMENTAL THEOREM OF ALGEBRA Since the equation consist of a positive degree of n, therefore it has at least one zero.

THE NUMBER OF ROOTS THEOREM

In the above equation the highest degree of exponent is 2, then there exist 2 roots of

P(x)

Descarte‟s rule of sign

Consider the change in the sign of the first term and second term.

last term.

( ) There are 2 positive roots taken from the original equation

FACTOR THEOREM ( )( ) ?

( ) ( )

40-77+28=0 16-44+28=0

-28+28=0 -28+28=0

0=0 0=0

Then, 4 and 7 are the roots of

NAME: _________________________________________ SCORE_____________

DATE: _________________

EXERCISES 1.5 FINDING THE ROOTS OF A POLYNOMIAL

using the different Polynomial Theorems Example

P(+) = 2 roots factor( x-7)(x-4)=0 then 4 and 7 are the roots.

P(-) 0

#of roots positive negative factors roots

1.

2.

3. –

4.

5.

6.

7.

8.

9.

10.

11.If Is a factor of the given

equation?

12.Find the value of k such that when

13.Find the value of k such that when e

14.If (x+1) is a factor of ____.

15.Find the value of k such that when

( )

MODULE 5: EXPONENTIAL AND LOGARITHMIC EQUATION

PRE- REQUISITE TOPICS

LAWS OF RATIONAL EXPONENT FACTORING

SOLVING POLYNOMIAL EQUATION

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

EXPONENTIAL FUNCTIONS

If b is any positive number, then the expression bx

=designates exactly one real

number for every real value of x. where b is no equal to zero.

f(x)= bx

Properties of

Logarithms

1. Product rule:

2. Quotient rule: (

)

3. Power rule:

4. If and and = x

x -2 -1 0 1 2

y 0.25 0.5 1 2 4

f(x)=

2x

y

-2

-1

0

1

2

NAME: _________________________________________ SCORE_____________

DATE: _________________

EXERCIES 1.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS

DIRECTIONS: Solve for x in the following equation below. Match Column A with the

correct value of x in

Column B. If the answer is not found in the selection, write the correct

answer.

Note: Calculator is allowed.

COLUMN A COLUMN B

1. A. x=4 P. x = 49/4

2. B. x= 5 Q. x = -8/3

3. C. x = -1/2 R. x = 6

4. ( ) D. x = 3 S. x = 0

5. E. x = 3/2 T. x = 4/49

6. F. x = ½

7. G.

8. H. x = 81

9. I. x = 16/9

10. J. x = 5/2

11.

K. x = 19/6

12.

L. x = -1

13. M. x = 2/21

14.

N. x = 21/2

15. O. √

B. Directions: Solve each equation applying the different properties of Logarithm.

16.

17.

18.

19.

20. (

)

21. =

22.

23.

24.

25.

C. EXPONENTIAL EQUATIONS LEADING TO QUADRATIC EQUATION

26. ( )

27. ( )

28. ( ) ( )

29. ( )

30. (

) (

)

31. = 4y

32. 33. 34.

35.

INTEGER PROBLEM

Let x, x+1, x+2, x+3, x+4,… for a consecutive integer

Let x,x+2, x+4, x+6, x+8,… for a consecutive even/ odd integer

Age Problem

Past & ago implies subtraction from the present age.

Ex. A is one year younger than B means A‟s age is x-1 and B‟s age is x.

In nth

year means that one needs to add the particular number to the

present age. In a given problem sentence, when one sees the word “is” this implies

equal.

Mixture

The word” is/ must be added to “means add the two mixtures.

The word “must be reduced to” means deduct the amount of mixture from

the other.

The word “to make a mixture of” or “to have a result of” means add up the

two mixtures.

Ex.

A 5 gram of 20% SALT SOLUTION must be added to a pure concentration of SALT solution must be added to make a 75% Salt solution.

5g( 20%) + x g ( 100%) = 75%(5g + x g)

WORK PROBLEM

SAME JOB TO BE DONE

X+Y = 1

RATE OF WORK= 1/X

MOTION PROBLEM

DISTANCE = RATE X TIME or d = rt

Over take problem d1 = d2

Opposite Direction

d1+ d2= dt

d1-d2= difference on distance

Worded Problems

1. Trixie has 10 pieces of P100- bills and 35 pieces of P20-bills. How much money does

she have?

2. Mollie has a total of P 4,800 consisting of P50 and P100-bills. The number of P50-

bills is 16 less than twice the number of P 100-bills. How many P50-bills does she

have?

3. Jonas has a jar in his office that contains 39 coins. Some are 5 cents and the rest

are 10 cents. If the total value of the coins is P2.55, how many 5 cents does he

have?

4. In problem number 3, how many 10 cents does Jonas have in the jar?

5. Erick has a box of coins. The box currently contains 40 coins, consisting of 5 cents,

10 cents, and 25 cents. The number of 5 cents is equal to the number of 25 cents,

and the total value is P5.80 How many of 25 cents of coin does he have in the box?

6. Ruth has P1900 consisting of P50 and P10 bills. The number of P10 bills is 5 less

than the number of P 50 bills. How many P10 bill does she have?

7. Mildred can sew a dress in ten days. What part of the dress is finished after 6 days?

8. Glen is 3 years older than his brother. Three years ago, Four years from now, the

sum of their ages will be 33 years, how old are they now?

9. Mr. Sta. Maria is five years older than his wife. Five years ago, his age was 4/3 her

age. What will be their ages 8 years from now?

10.A man can wash the car in 120 minutes if he works alone. His son, working alone

can do the same job in 3 hours. How long will it take to take them to wash the car if

they work together?

11. An inlet pipe can fill a tank in 9 minutes. A drain pipe can empty the tank in ten

minutes. If the tank is empty and both pipes are open, how long will it take before

the tank overflows?

12. Find three consecutive even integers such that four times the first less the

third is six more than twice the second.

13. Find three consecutive integers such that the sum of the first plus one-third

of the second plus three eights of the third is 25.

14. A paint that contains 21% green dye is mixed with a paint that contains

15% green dye. How many gallons of each must be used to make 60 gallons

of paint that is 19% green dye.

15. A chemist has 10 milliliters of a solution that contains a 30% solution of

acid. How many milliliters of pure acid must be added in order to increase the

concentration to 50%?

1. ( )

2. ( )

3. ( ) ( )

4. ( )

5. (

) (

)

6. = 4y

7. 8. 9.

10.