lecture 6 - linear systems 2 Flashcards
What is rank?
Notation is rho
highest order of non-singular submatrix.
what is the rank for a null matrix?
0
What is the rank for a singular matrix?
not 3, test for whether its 1 or 2.
e.g.
1 1 1
2 2 2
3 3 3 is rank of 1
but
1 -2 0
2 2 6
0 1 1 is rank 2
What is a determined matrix?
has only 1 solution.
what is an undetermined matrix?
Infinite solutions
What is a compatible matrix?
has at least 1 solution
what is an incompatible matrix?
has no solution
What is the rouche-capelli theorem?
The system is compatible IF AND ONLY IF both A and A(c) (complete matrix) have the SAME RANK! so
Rho(A) = Rho(A(c)) = Rho
according to the rouche capelli theorem when is a matrix incompatible?
when rank of A is smaller than rank of A(c).
No solutions.
Always check if this is the case for non-homogenous systems.
When should u use cramers method to solve matrixes in rouche capelli theorem?
if both ranks of A and A(c) are the same and rank = n = m OR rank = n < m.
n = rows
m = columns
When is a matrix undetermined?
if rank is < n and rank <= m then there would be a total of infinity^(n - rank) solutions.
How many undefined parameters will there be for infinity^1
infinity^2
infinity^1 = 1 undefined parameter (make something alpha)
infinity^2 = 2 undefined parameters (make something alpha and beta)
How to get a solution for an undetermined matrix?
- find the number of solutions using the no. of rows and rank of the matrix
- cancel any rows if possible. Make sure not to change the rank.
- set arbitrary values to alpha/beta and find values for x,y,z with substitution
How to cancel if matrix in form:
R1 + R2 = R3
Does not matter which is cancelled, but only 1. Rank will be 1 as the system is singular.
What is a homogenous system?
if all known vectors are null
1 1 1 1 | 0
7 2 3 2 | 0
1 3 1 1 | 0
-1 2 4 3 | 0
