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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
Hey vanmaiden.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
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.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.
Well in terms of independence, I refer to linear independence. Two things are linearly independent if one can not be expressed as a linear combination of other things. When they are linearly dependent, we can express something in terms of a linear combination of the other things.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.
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?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.
You can pick the pivot variable if you want to, it doesn't really matter but it would make sense at least for consistency.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?
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?You can pick the pivot variable if you want to, it doesn't really matter but it would make sense at least for consistency.
Sorry, when I said dependent variables I meant like in the situation where we had x + y = 0 where one variable is independent and the other is dependent.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?