What is a pivot column in a system of linear equations?

[number0]

Homework Statement

What is a pivot column?

Homework Equations

The Attempt at a Solution

I tried google for some examples but I cannot find any. I need to know what exactly a pivot column is because I am trying to do a Linear Algebra problem about consistency and inconsistency. Can anyone provide me some examples of what a pivot column is (or is not) on the last column of a given matrix?

Thanks.

 

[Deveno]

consider the row-echelon matrix A =

[1 0 2 3]
[0 1 1 4]
[0 0 0 1]
[0 0 0 0]

the "pivot columns" are the columns that contain the leading 1's of the rows. in this example, they are columns 1,2 and 4. "non-pivot columns" are linearly dependent on preceding ones. for example column 3 is twice column one plus column two.

pivot columns are important, because they form a basis for the column space, which has dimension = rank(A). the number of pivot columns in an mxn matrix is always equal to the number of non-zero rows in a row-reduced matrix.

the concept only applies to row-reduced matrices (row-reduction is how we reveal which columns are independent). but having row-reduced a matrix, we can use the pivot columns of the reduced form, to identify a basis for the column space of the original matrix, by choosing the same columns as basis vectors. in our example, if we had started with some other matrix, B, and row-reduced it to A, the 1st, 2nd and 4th columns of B would form a basis for the column space of B (also known as the range, or image of B).

 

[number0]

consider the row-echelon matrix A =

[1 0 2 3]
[0 1 1 4]
[0 0 0 1]
[0 0 0 0]

the "pivot columns" are the columns that contain the leading 1's of the rows. in this example, they are columns 1,2 and 4. "non-pivot columns" are linearly dependent on preceding ones. for example column 3 is twice column one plus column two.

pivot columns are important, because they form a basis for the column space, which has dimension = rank(A). the number of pivot columns in an mxn matrix is always equal to the number of non-zero rows in a row-reduced matrix.

the concept only applies to row-reduced matrices (row-reduction is how we reveal which columns are independent). but having row-reduced a matrix, we can use the pivot columns of the reduced form, to identify a basis for the column space of the original matrix, by choosing the same columns as basis vectors. in our example, if we had started with some other matrix, B, and row-reduced it to A, the 1st, 2nd and 4th columns of B would form a basis for the column space of B (also known as the range, or image of B).

Thanks for the explanation. However, if the matrix were to be defined as

[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1] .

Would the last column still be considered a pivot column? If not, what is the reasoning behind this?

 

[Stephen Tashi]

You have to state more of the context in which the phrase "pivot column" was used, otherwise it could mean a hundred different things.

In general terms, there are various algorithms for solving problems such as solving systems of equations, solving linear programming problems etc. Many of these algorithms proceed in steps that involve picking a particular element in a matix and then doing something with the row or column that contains that element. The matrix entry that is picked is called the "pivot" and it's column is, of course, the " pivot column".

What criteria is used to pick the "pivot" and thus select the "pivot column" depends on what algorithm you are discussing.

 

[number0]

The context is "solving a system of linear equations," where I am referring to the column before the last column in an augmented matrix. For example, the bolded column in the following matrix:

[1 0 0 0 | 3]
[0 1 0 0 | 2]
[0 0 1 0 | 1]
[0 0 0 1 | 6].

 

[Stephen Tashi]

The context is "solving a system of linear equations," where I am referring to the column before the last column in an augmented matrix.

There are several algorithms for solving a system of linear equations and several matrices appear as these algorithms progress. So I can't answer the question "what is a pivot column" from that description.

 

[number0]

There are several algorithms for solving a system of linear equations and several matrices appear as these algorithms progress. So I can't answer the question "what is a pivot column" from that description.

I am trying to find whether if the "system of linear equations" is consistent or not. This can be be measured by looking at the column before the augmented column. If it is not consistent, then the column before the augmented column is a pivot column. Otherwise, it is not.

 

[Stephen Tashi]

From what you have said, all I can guess is that your textbook intends that you perform some operation with a column and that it has told you a rule for selecting the "pivot column". The fact that it used the term "pivot column" does not reveal what it intends for you to do with the column.

That's the best I can do without knowing what specific algorithm (and I mean "algorithm" not "topic") you are studying. You haven't even quoted the sentence in your material that uses the term "pivot column".

 

[Deveno]

I am trying to find whether if the "system of linear equations" is consistent or not. This can be be measured by looking at the column before the augmented column. If it is not consistent, then the column before the augmented column is a pivot column. Otherwise, it is not.

i believe you are mistaken, a system is inconsistent if the augmented column is a pivot column.