1.2 Row Reduction and Echelon Forms Flashcards
What is a non-zero row?
A row with at least one non-zero
What is a leading entry?
The first non-zero number in a row
Which three properties prove that a rectangular matrix is in row echelon form?
- All non-zero rows are above all zero rows
- Each leading entry of a row is in a column to the right of the leading entry in the row above it
- All entries in a column below a leading entry are zeros
what are the properties of a rectangular matrix in reduced echelon form?
The same three properties for row echelon form
- Leading entry in each non-zero row is one
- Each leading one is the only non-zero entry in its column
What is echelon matrix or reduced echelon matrix?
A matrix in echelon form
What does row reduced mean?
Any nonzero matrix may be reduced by elementary row operations into more than one matrix in echelon form
What does an echelon form of A mean?
If matrix A is reduced to an echelon form say U
What does the abbreviation RREF stand for?
Reduced row echelon form
What are pivot positions?
Location in matrix with leading one in reduced echelon form
What are pivot columns?
A column with a leading one
What is a pivot?
A non-zero number in a pivot position that is used as needed to create zeros via row operations
What does backward phase mean?
Reducing a matrix to reduce echelon form
What does the forward phase mean?
Reducing a matrix to echelon form
What does partial pivot mean?
When our computer program usually selects pivot with the largest absolute value
What are basic variables?
The variable that corresponds to a pivot column in a coefficient matrix
What are free variables?
Other variables that do not correspond to pivot columns
What is a flop?
An arithmetic operation on two real floating point numbers
What are parametric descriptions?
A form where free variables act as parameters
What does theorem one say about the uniqueness of reduced row echelon form?
Each matrix is equivalent to one and only one reduced echelon matrix
What is theorem two?
System is consistent if right most column of the matrix is a pivot column
What are the steps of using row reduction to solve a linear system?
- Write an augmented matrix of the system
- Use a reduction algorithm to get an equal matrix in echelon form, decide whether the system is consistent, if there are no solutions stop
- Continue row reduction to get reduced echelon form
- write the system of equation to matrix achieved in step three
- Rewrite each non-zero equation so basic variables are expressed in terms of any free variables
