QUESTIONS - CDOG - LINEAR ALGEBRA

QUESTIONS - ANSWERS -CDOG -LINEAR ALGEBRA

On 11/19/06, cdog wrote:
> cdog has left a new comment on your post "QUESTIONS - MAXIMA MINIMA":
>
> QUESTION ABOUT LINEAR DEPENDENCE AND LINEAR INDEPENDENCE. Can someone please
> explain to me (like I am a 5-year old) the basic difference between linear
> independence vs. linear dependence. Perhaps, I'm reading my Linear Algebra
> text incorrectly, but they appear to be a contradiction in terms. A set is
> dependent if the determinent of the coefficient matrix=0? Otherwise, the set
> is independent? Is this a correct statement?
WELL AS DESIRED BY YOU, THIS ASPECT STARTING WITH SIMPLE EXAMPLES WAS POSTED UNDER MATHS-6.EQUATIONS IN OCT '06,WHICH IS REPRODUCED BELOW.
QUOTE:
Simultaneous equations involving 2 unknowns - x and y are often referred by school students for different methods of solution and more importantly to find whether they are consistent, independent or inconsistent. Let us see 3 simple examples and elaborate on this particular aspect of existence and uniqueness of solution.

1. x+y =2 and x-y =0
2. x+y =2 and 2x+2y = 4
3. x+y =2 and 2x+2y =3

We adopt 2 methods to determine this aspect.

1.Algebraic method

2.Graphic method


I.Independent equations

Test 1 . Algebraic method.

Find the ratios of coefficients of unknowns in the 2 given equations

In ex.1…it is

1/1 =1 for x

and

1/-1 =-1 for y.

the ratios are not equal

IF THE RATIOS ARE DIFFERENT,THEY ARE INDEPENDENT EQUATIONS AND WE GET UNIQUE SOLUTION.

Here we find that x = y = 1 is the one and only one solution.

In ex.2….it is

1 / 2 for x

and

1 / 2 for y

the ratios are equal.

IF THE RATIOS ARE EQUAL THEN THEY ARE DEPENDENT EQUATIONS AND WE CAN HAVE TWO POSSIBILITIES.

A. We may have infinite solutions

We test for this possibility by finding the ratio of constant terms. Here the ratio is 2 / 4 = 1 / 2 which is same as the ratio of coefficients of unknowns.

IF RATIO OF COEFFICIENTS OF UNKNOWNS IS SAME AS THE RATIO OF CONSTANT TERMS THEN THEY ARE DEPENDENT / CONSISTENT AND WE HAVE INFINITE SOLUTIONS.

Here we have x = y = 1 or x = 0 and y =2 or x=2 and y=0 etc..are all possible solutions. In-fact any set of values x = x and y = 2 – x is a solution.

PLEASE NOTE THE COMMENTS UNDER EQUATIONS AND IDENTITIES IN THIS CONTEXT

B. We may have no solution.

IF RATIO OF COEFFICIENTS OF UNKNOWNS IS NOT SAME AS THE RATIO OF CONSTANT TERMS THEN THEY ARE CALLED INCONSISTENT AND WE HAVE NO SOLUTION.

In ex.3..it is

1 / 2 for x

1 / 2 for y

and

2 / 3 for constants which is not equal to the above ratio.

Hence the equations are in-consistent and we have no solution.

Test 2…Graphic method.

Draw graphs of the 2 equations on the same scale on the same paper.

If the 2 graphs intersect , then the point of intersection is the solution.

In Ex.1 we find that the 2 graphs…straight lines in fact intersect
at x = y = 1.Hence they are independent equations and the point of intersection viz.x = y = 1 is the solution.

In Ex.2 …we find that the 2 graphs ….straight lines in fact are coincident. Hence they are dependent/consistent and any point on the line is a solution. Say x = y = 1 or…x=2 and y =0 etc..

In Ex.3...we find that the 2 graphs...straight lines in-fact are parallel and do not intersect.hence, they are inconsistent equations and there is no solution.
UNQUOTE:
NOW COMING TO YOUR POINT OF DETERMINANT OF COEFFICIENT MATRIX BEING ZERO OR MORE PRECISELY AS YOU WOULD KNOW LATER,ITS RANK BEING LESS THAN ITS ORDER.
IN THE ABOVE EXAMPLES,TO EXPLAIN TO SCHOOL STUDENTS,WE TERMED IT AS RATIO OF COEFFICIENTS.DETERMINANTS OF THE 3 COEFFICIENT MATRICES IN THE ABOVE EXAMPLES ARE,
EX.1
|1, 1|
|1,-1| = 1*(-1)-(1*1)=-2
EX.2
|1,1|
|2,2| = 1*2 - 1*2 =0
EX.3
|1,1|
|2,2| = 1*2 - 1*2 =0

SO THE RULE IS CORRECT
IF THE DETERMINANT OF COEFFICIENT MATRIX IS ZERO,THEN THE EQNS. ARE TERMED DEPENDENT
BUT WHETHER THEY ARE CONSISTENT OR NOT DEPENDS ON THE CONSTANT TERMS AS BROUGHT OUT BY ME IN THE ABOVE LESSON. THE TRANSLATION OF THIS TO MATRICES,YOU WILL LEARN SOON..
A FURTHER GENERAL DEFINITION FOR MORE THAN 2 EQUATIONS CAN BE AS FOLLOWS.
IF WE CAN EXPRESS ANY ONE EQUATION AS A LINEAR COMINATION OF OTHER EQUATIONS ,THEN THEY ARE SAID TO BE LINEARLY DEPENDENT.
THAT IS SUPPOSE WE HAVE SEVERAL EQNS.LIKE
A1X+B1Y+C1Z+....ETC ...= 0..........1
A2X+B2Y+C2Z+....ETC... = 0.............2
A3X+B3Y+C3Z+....ETC... = 0...................3
-----------------------------------------------------------------------
-----------------------------------ETC-------------------------------
AND WE CAN WRITE .....
K1[A1X+B1Y+C1Z+....ETC..]+K2[2X+B2Y+C2Z+....ETC.]+K2[A3X+B3Y+C3Z+....ETC..] =0........................WITH NOT ALL K1,K2K3 ETC..ZERO ,
THEN WE SAY THEY ARE LINEARLY DEPENDENT.
IF WE CANNOT HAVE THE SUM AS ZERO UNLESS K1=K2=K3=...ETC=0 ALL , THEN WE SAY THEY ARE LINEARLY INDEPENDENT.