1.2 Gaussian Elimination Flashcards
According to Anton, what is the “general solution” to a linear system?
If a linear system has infinitely many solutions, then a set of parametric equations from which all solutions can be obtained is called a general solution of the system.
The variables corresponding to the leading 1’s in a reduced row echelon augmented matrix, we call the __. The remaining variables are called __.
Corresponding to the leading 1’s are the leading variables.
The remaining variables are the free variables.
The algorithm to produce a row echelon form is called…
Gaussian elimination
The algorithm to produce a reduced row echelon form is called…
Gauss–Jordan elimination
A system of linear equations is said to be _ if the constant terms are all zero.
homogeneous
Every homogeneous system of linear equations is…
…is consistent, because they all have at least the trivial solution.
Solutions to homogenous linear systems that are not all 0’s are called…
…nontrivial solutions (infinitely many).
What solution possibilities are there for homogeneous linear system?
Either it has only the trivial solution, or it has infinitely many (non-trivial) solutions in addition to the trivial solution.
When is a homogenous system assured of having nontrivial solutions?
Whenever the system involves more unknowns than equations, it will have infinitely many solutions.
What can we assert when a homogenous linear system has more unknowns than equations?
It is guaranteed to have non-trivial solutions (infinitely many).
Elementary row operations can not alter…
…columns of zeros.
What is the free variable theorem?
If a homogeneous linear system has n unknowns, and it’s reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n − r free variables.
