(compare the system (2.1.2) with the original system.)
(compare the system (2.1.3) with (2.1.2) or the system (2.1.4) with (2.1.3).)
So, in Example 2.1.4, the application of a finite number of elementary operations helped us to obtain a simpler system whose solution can be obtained directly. That is, after applying a finite number of elementary operations, a simpler linear system is obtained which can be easily solved. Note that the three elementary operations defined above, have corresponding INVERSE operations, namely,
It will be a useful exercise for the reader to IDENTIFY THE INVERSE OPERATIONS at each step in Example 2.1.4.
The linear systems at each step in Example 2.1.4 are equivalent to each other and also to the original linear system.
In this case, the systems and vary only in the equation. Let be a solution of the linear system Then substituting for 's in place of 's in the and equations, we get
Therefore,
Use a similar argument to show that if is a solution of the linear system then it is also a solution of the linear system
Hence, we have the proof in this case. height6pt width 6pt depth 0pt
Lemma 2.2.4 is now used as an induction step to prove the main result of this section (Theorem 2.2.5).
If Lemma 2.2.4 answers the question. If assume that the theorem is true for Now, suppose Apply the Lemma 2.2.4 again at the ``last step" (that is, at the step from the step) to get the required result using induction. height6pt width 6pt depth 0pt
Let us formalise the above section which led to Theorem 2.2.5. For solving a linear system of equations, we applied elementary operations to equations. It is observed that in performing the elementary operations, the calculations were made on the COEFFICIENTS (numbers). The variables and the sign of equality (that is, ) are not disturbed. Therefore, in place of looking at the system of equations as a whole, we just need to work with the coefficients. These coefficients when arranged in a rectangular array gives us the augmented matrix
Whereas the matrix is not row equivalent to the matrix