Reduced row echelon form: Detailed information on echelon form
Opening a discussion regarding reduced row echelon form, we should divide this term called the reduced row echelon form into parts and concentrate mainly on the echelon form for the first place. This way we will grasp the concept of the term reduced row echelon form better and easier. From now on you will know that echelon form means that our matrix is located in one of the two stages; either in row echelon form or in reduced row echelon form. This fact means that the matrix should satisfy the requirements we are going to present you below. Look at the example of the matrix in echelon form and go on reading not to complain anymore that you know nothing about the reduced row echelon form!
That’s it! Now your knowledge on Gaussian Elimination and reduced row echelon form is even firmer and more reliable!
As mentioned previously in this article, any matrix you will ever see can be transformed into reduced row echelon form through already discussed Gaussian Elimination. Did you know that most of the graphic calculators have a function enabling you to transform a specific matrix into reduced row echelon form? If you have decided to get reduced row echelon form manually then apart from considering the requirements that reduced row echelon form has, you also need to have a knowledge on how to interchange one row with another one, how to multiply one row by a non-zero constant and finally what is the right way of replacing one row with: one row, plus a constant, times quite another row. In addition, it should be noted then when dealing with reduced row echelon form knowing the rules is not enough for solving a task; you also need to make sensible and logical actions. If you do, then be sure that the topic of the reduced row echelon form will seem to you simple, interesting and challenging in a good way!
In this section of the article, we would like to discuss reduced row echelon form and general solutions. Suppose we have a matrix of a system Ax = b which is simplified to the below mentioned reduced row echelon form.
12003
[ 00104 ]
00015
The people interested in reduced row echelon form have probably already guessed that the simplified system will be:
- x1+ 2×2 = 3
- x3 = 4
- x4 = 5
Now it is easier to give a general solution:
- x1 = 3 – 2×2
- x3 = 4
- x4 = 5
Please, note that here x2 is arbitrary!
It is worth mentioning that the coefficients in the general solution are vividly presented in the reduced row echelon form. Thus, we can claim with confidence that the general solution of Ax=2 absolutely corresponds to the reduced row echelon form of [A b]. As you can see from the above-mentioned information and examples on reduced row echelon form, nothing is too difficult. Of course, if you previously had absolutely no idea about reduced row echelon form then we will agree that starting the process of learning, reduced row echelon form may be a bit challenging for you. However, reading articles and essays on such interesting topic as reduced row echelon form may evoke fire in your soul and brain inspiring you to know more and more in future!
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