Line stairs form

When a certain line becomes the shape to be provided as a result of method of elimination of the gauss in a field of the linear algebra of the mathematics, it is said that the line is stairs form (かいだんけい British: echelon form). Line stairs form (row echelon form) is the stairs form where method of elimination of the gauss is provided for the line of the line when it is acted, and the line stairs form (column echelon form) is defined equally, too. A sufficient condition for a certain line to be line stairs form is that the dislocation line is line stairs form. Therefore, I understand that it is enough if I consider only line stairs form in the following. Similar property for the line stairs form is easily provided by performing translocation of a line of all line stairs form to treat.

I say time when next is established to be concrete when a line is line stairs form:

• A line (the line where at least one ingredient is not zero) having the ingredient which is not zero is located on the top than a line having only zero to an ingredient (if the line only as for the zero ingredient exists, they are located in the bottom of the line).
• A chief ingredient (the ingredient which is zero in the left least of the line is called pivot (English version)) is located the right side truly than the chief ingredient of the line on the line. (^[1] that the textbook said to have to be 1 has the chief ingredient by all means)

From two conditions mentioned above, I understand that all the ingredients below the chief ingredient of a certain line are zero [²].

In an example of the line stairs form of 3*5 line, shown below:

${\displaystyle \left[{\begin{array}{ccccc}1&a_{0}&a_{1}&a_{2}&a_{3}\\0&0&2&a_{4}&a_{5}\\0&0&0&1&a_{6}\end{array}}\right]}$

Table of contents

Line conciseness stairs form

I say that I meet a condition to give below that a line is line conciseness stairs form (called the line canonical form). I can calculate the line conciseness stairs form of a certain line by method of elimination of the gauss. However, unlike line stairs form, the line conciseness stairs form is not a thing depending on the calculation method in one idea.

• All the lines having the ingredient which is not zero are located on the line that only zero has toward an ingredient.
• Chief ingredients are always located the right side truly than the chief ingredient of the upper line.
• All chief ingredients are 1 and are the ingredient which is not only horn zero in the line including the chief ingredient [³].

The line that is line conciseness stairs form meets all conditions about the line stairs form and is limited more.

I raise example of the line that is line conciseness stairs form next:

${\displaystyle \left[{\begin{array}{ccccc}1&0&0&0&b_{1}\\0&1&0&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}$ 

Here, in the left side of the line, it should be always noted that it is not an identity matrix. For example, the line to give next is line conciseness stairs form:

${\displaystyle \left[{\begin{array}{cccccc}1&0&0&1/2&0&b_{1}\\0&0&1&-1/3&0&b_{2}\\0&0&0&0&1&b_{3}\end{array}}\right]}$ 

An L meat standard form for the lines of the integer coefficient is the line stairs form which are calculated without introducing any rational number and denominator using Euclid division. On the other hand, the line conciseness stairs form of the line of the integer coefficient includes the ingredient of the non-integer generally.

Conversion to line stairs form

I can convert any line into line stairs form by performing line basics transformation called the method of elimination of the gauss in a limited time. As for the line basics transformation, the line space of the line stairs form becomes equal to the line space of the original line to save the line space of the line.

As a result, there is not the stairs form provided in the one idea. For example, doubling it is line stairs form the arbitrary scalar of the line that is line stairs form. However, the line "conciseness" stairs form is one idea for all lines. As for this, the line of non-zero of the line conciseness stairs form means that it is only horn line conciseness stairs generation meeting for the line space of the original line.

Coalition linear equation

When a coalition linear equation is line stairs form, I say that the extended coefficient line is line stairs form. Similarly, I say that the extended coefficient line is line conciseness stairs form when a coalition linear equation is line conciseness stairs form or a standard form.

I can think that the canonical form gives a solution of the linear equation system concretely. Actually, a necessary and sufficient condition of 為 where equation system does not have a solution toward (inconsistent) is that the equation that one line of the canonical form expresses is written with 1 = 0. The variable moved to the right side becomes arbitrary, and the variable corresponding to the chief ingredient is expressed for the fixed number or a linear function (if there is a variable moved like a perilla in the right side but) of the variables moved to the right side if I transpose all the clauses not to support a chief ingredient in the right side of the equation when there is not such a line.

Explanatory note

1. ^ See, for instance, Larson and Hostetler, Precalculus, 7th edition.
2. ^ Meyer 2000, p. 44
3. ^ Meyer 2000, p. 48

References

• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0 - 89,871-454-8, http://www.matrixanalysis.com/.

Outside link

Wiki books have a Row Reduction and Echelon Forms-related commentary book, a textbook.

• Interactive Row Echelon Form with rational output

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