- #1
vanmaiden
- 102
- 1
Why do we solve for only the pivot variables when we are trying to solve a system of equations using reduced row echelon form?
Thank you
Thank you
vanmaiden said:Why do we solve for only the pivot variables when we are trying to solve a system of equations using reduced row echelon form?
Thank you
chiro said:Hey vanmaiden.
The simple explanation is that the pivot variables end up telling us what is independent and what is not.
If we however take a general matrix and reduce it and find that it has a pivot count less than or equal to the number of rows, then it means that the vectors are linearly dependent and the basis for the space can be written in terms of a lesser number of vectors.
vanmaiden said:I'm still somewhat confused, but not as much as I was before. I can now see the presence of standard basis vectors when we put a matrix into reduced row echelon form, but I don't see how their independence from the rest of the vectors make their variables the ones that we solve for. I guess what I'm trying to say is I see the word "independence," but I don't know how to connect the independence of a vector to the independence of a variable.
chiro said:This translates into x + 2y + 3z + 4w = 0. This means that we pick one independent scalar variable and three dependent scalar variables.
since x + 2y + 3z = 0, pick one independent, the others are dependent.
vanmaiden said:Do we pick the pivot variable (in that case, "x") because it seems the easiest to use when finding the solution due to its independence from the rest of the column vectors?
chiro said:You can pick the pivot variable if you want to, it doesn't really matter but it would make sense at least for consistency.
vanmaiden said:I just find it somewhat strange that you can choose which variable is the independent variable and which one is the dependent variable after putting a matrix in RREF. I mean it seems like such a choice would be more defined and rigid rather than at one's whim. Am I viewing this correctly?
Pivot variables in reduced row echelon form refer to the columns of a matrix that contain the leading entry (a non-zero number) in each row. These variables represent the basic variables in a system of linear equations and can be used to find a unique solution.
To identify pivot variables, look for the columns with the leading entry in each row. These columns will contain the pivot variables. In reduced row echelon form, these variables will have a value of 1, while all other entries in the column will be 0.
Pivot variables are important because they represent the basic variables in a system of linear equations. They provide a way to find a unique solution and can help to simplify the solution process.
To solve a system of linear equations, use the pivot variables to set up a back substitution process. Start with the bottom row and use the value of the pivot variable to eliminate the other variables in that row. Then, move up to the next row and use the values of the pivot variables in that row to eliminate the remaining variables. Continue this process until you reach the top row, where you will have a unique solution.
No, a system of linear equations can only have one pivot variable per row. This is because the leading entry in each row must be a unique and non-zero number. If there were more than one pivot variable in a row, it would indicate that the system is inconsistent and has no solution.