Row Reduction Quiz Flashcards
Consistent
A SOE of eqs. that has @ least one solution
Inconsistent
The solution set is empty
Linear equation
An equation where the highest power of variables is one
Standard Form
All variables on one side set equal to 0
System of linear equations
A set of eqs. we want to be true @ the same time
Solution
A set of all values which satisfies all eqs. in the system
Solution set
A system of all solutions; can have one, many, or more
Square Matrix
A matrix with equal columns and rows
Adjoined
Matching a matrix with a vector
3 x 4
3 is the height 4 is the width
rows can be written as…
a coefficient matrix
The answer can be written as…
vector
Augmented Matrix
Combination of a coeff. Matrix and answers vector
Replacement
We can add to a row any multiple of another row
Scaling
Multiply a row by a (non-zero) constant
Re-ordering
Swap the order of two rows
The first step of the Algorithm
Make top left entry = 1 by scaling or re-ordering
The second step of the Algorithm
Use elimination or replacement to make all entries below that 1 into 0.
The third step of the Algorithm.
Make the left-most non-zero entry m row equal to one (creating a pivot)
The 4th step of the algorithm
Use the new pivot to eliminate all values below
The 5th step of the algorithm
Repeat the 3rd step until the matrix is in echelon form
What is the 6th step of the algorithm?
After reaching echelon form, work up and left
What is the 7th step of the algorithm?
Use the pivots to replace all values above the one
Reduced Echelon Form
All values below and above the pivots are equal to 0
The equation isn’t independent
A whole row of zeros in the matrix
Not every row has a pivot but we are already in reduced echelon form, where the pivot should be the…
Free Variable
How to solve with a free variable
- Take out of the matrix and write as equations
- Put all free variables to one side
- Those equations then become the solution set
Dimension
of a solution set is the # of free variables
Unique Solution
If the matrix is consistent and there is a pivot in every column of the coeff matrix
Infinite Solutions
When one of the columns of the coeff. matrix does not have a pivot or when the whole row doesn’t make sense to the answer
Plane
A linear equation in 3 dimensions
