Chapter 1.2 Flashcards
The property a matrix in row echelon form has to have to be in reduced row echelon form.
Each column that contains a leading 1 has zeros everywhere else in that column.
The three properties of a matrix in row echelon form
1) The first nonzero number in each row is a 1, the leading 1.
2) Rows with all zeros are grouped on the bottom of the matrix.
3) In two succesive rows, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.
Leading variables
The variables that correspond to the leading 1s
Free variables
The variables used to solve the leading variables.
Def: A general solution
A set of parametric equations from which all solutions (to a linear system with infinitely many solutions) can be obtained by assigning numerical values to the parameters.
A homogeneous linear system
A linear system where the constant terms are all zero
The trivial solution to a homogeneous linear system
Setting all variables to zero
The non-trivial solution to a homogeneous linear system
Performing a gauss-jordan elimination and solving the leading variables.
The Free Variable Theorem
“If a homogeneous linear system has n unknowns and if the reduced row echelon form of it’s augumented matrix has r nonzero rows, then the system has n - r free variables.”
How many solutions does a homogeneous system with more unknowns than equations have and why?
Infinately many because at least one unknown will have to be a free variable.
What are the three steps of back substitution?
1) Solve the equations for the leading variables.
2) Beginning with the bottom equation and working upward, succesively substitute each equation into all the equations above it.
3) Assign arbitrary values for the free variables if any.
The 5-step iteration of Gaussian Elimination
1) Locate the left most column that does not consist entirely of zeros
2) Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column found in step 1.
3) If the entry that is now at the top of the column found in step 1 is a, multiply the first row by 1/a in order to introduce a leading 1.
4) Add suitable multiples of the top row to the rows below so that all entries below the leading 1 become zeros.
5) Now cover the top row and begin again by step 1.
The final procedure to turn a Gaussian Elimination into a Gauss-Jordan Elimination.
Begin with the last nonzero row and working uppward, add suitable multiples of each row to the rows above to introduce zeros above the leading 1s.
