Basic variable -- variables in a pivot column in reduced echelon form
Free variable -- any non-basic variable
Find the general solution:
⎝⎛100600220−5−80−2−11−437⎠⎞R2+R3→⎝⎛100600220−5−80−201−4107⎠⎞R1+2R3→⎝⎛100600220−5−8000110107⎠⎞R2∗21→⎝⎛100600210−5−400011057⎠⎞R1−2R2→⎝⎛1006000103−40001057⎠⎞
⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧x1=−6x2−3x4x2 is freex3=5+4x4x4 is freex5=7
A linear system is consistent if and only if the rightmost column of the reduced echelon matrix is not a pivot column.
If consistent: There's only a unique solution if there's no free variables. (Free variables imply infinite solutions)
- "A matrix can be reduced to different reduced echelon forms." (FALSE)
- "A matrix can be reduced to different echelon forms." (TRUE)
- "Row reduction only applies to augmented matrices." (FALSE)
- "Basic variables correspond to pivot columns in the augmented matrix." (FALSE (only true for coefficient matrices))
- "If one row of the echelon form of an augmented matrix is [00050], then the solutions are consistent." (FALSE (can't determine without knowing other rows))
- "Whenever a system has free variables, the solution set is infinite." (FALSE (only true for consistent systems))
- "If two systems of linear equations have the same solution set, then the associated row reduced echelon forms of the augmented matrices are the same." (FALSE)