Linear Algebra - Purpose to Application

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Let's start with a problem, 
                                                 
                                                              X + Y = 3
                                                              X - Y = 1  ...............(a)

what are the methods to solve the linear equation of two variables -

1. Elimination method
2. Substitution method
3. Cross multiplication method
4. Graphical method

Using above methods it is easy to solve the equation (a) and the solution of the equation is X = 2,Y = 1.

Now solve,

                                                            U + V + X + Y + Z = 4
                                                          -U + V - X + Y + Z = 0
                                                           U - V - X + Y + Z = 0
                                                           -U - V - X + Y + Z = -2
                                                           U - V + X - Y + Z = 0  ...............(b)

solve the equation (b).

Answer is U = V = X = Y = 1, Z=0.

Question is the degree of difficulty lies in solving the equation (b). Let me guess some of the problem you might be face,

(Note - There are always some special methods/techniques to solve linear equations, I advise you to keep it side for now because that methods/techniques are used to solve particular linear equations, but we are discussing here the general solution that is useful for all)

1. If we use elimination method you have to do so many steps by the nature of the equation you may can get solution but not always.
2. Similarly for substitution method you have to start with first equation then have to substitute one by one as it's only 4 variables, it will stop after 3 time substitute but what about if we have more than 4 variables.
3. Cross multiplication itself is a complex method but useful but writing and cross-multiply each one not possible for variable more than 2 variables.
4. If we use graphical method it's complex to draw 5 dimensional picture, it's started difficult after dimension 3.

Here the role of Linear Algebra comes in. Although at first it's start with a simple solution to the above problem later that leads us to vast area of the subject.

Let's see how it's start. For that turn the equation (b) into another form that is,

                                                      AX = B ..............(c)

where 
Now it's turn to what we called matrices. Now we can solve it by,

                                                      AX = B
multiply both side A-1

                                                  =>A-1AX = A-1B
                                                  => X = A-1B
where 
So,
So we get the solution, U = V = X = Y = 1, Z=0 as desired.

Here is the importance of matrix, although depend on problem different methods are there but all start with AX=B that matrization of the problem. It is true that in higher variables calculating matrix, its inverse is lengthy but transfer the original equation in matrix make the problem much easier. 

Later blogs we will discuss different tactic to solve our one and only primary goal to solve linear equations of several variables and how linear algebra evolve to reach our mission.



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