Chapter 1 Flashcards
System
A collection of linear equations involving the same variables
Linear equation
Can be written in the form
ax+ay+az…..=b
Solution
A list of values that make a system true
Solution set
A system of all possible solutions.
How many solutions can a system set have
0, 1, of infinite.
Coefficient matrix
Records the numbers multiplied by the variables.
Augmented matrix
Records all numbers in the system
Elementary row operations
Replacement
Interchange
Scaling
Row equivalent
If one can be transformed into another by elementary row operations.
Leading entry
The first non zero entertaining in a row
Echelon form
1 leading entries are in left to right order
2 any all zero row is at the bottom
Reduced echelon form
Everything with echelon and
3 all leading entries are 1
4 all leading entries are the only non zeros in there columns
Pivot column
A column with a leading entry in echelon form
Basic variables
Correspond to a pivot column
Free variables
Do not correspond to a pivot column
Parametric solution
All basic variables are solved in terms of zero or more free variables.
Theorem 2: The existence and uniqueness theorem
1 the system is consistent iff the right-most-column of M is not a pivot column
2 if the system is consistent and all variables are basic variables, then there is only one solution
3 if the system is consistent and has at least one free variable then it has infinite solutions.
Column vector
A matrix that consists of only one column
Span
Set of all linear combinations.
Theorem 4 the following are either all true or all false
1 the equation Ax=b has a solution for every vector b in R^m
2 every vector b in R^m is a linear combination of the columns of A
3 the columns of A span R^m
4 a has a pivot position in every row.
