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Learning Goals
- Define Linear Algebraic Equations
- Solve Systems of Linear Equation by Hand
using
- Gaussian Elimination
- Cramer’s Method
- Distinguish between Equation System
Conditions: Exactly Determined,
Overdetermined, Underdetermined
- Use MATLAB to Solve Systems of Eqns
Cramer’s Method for Eqn Sys
- Solves equations using determinants.
- Gives insight into the
- existence and uniqueness of solutions
- Identifies SINGULAR (a.k.a. Divide by Zero) Systems
- effects of numerical inaccuracy
- Identifies ILL-CONDITIONED (a.k.a. Stiff) Systems
Cramer’s Method – Illustrated-
- Now, to Find The Individual Solns, Sub The Constraint Vector for the Variable Coefficients and Compute the Determinant for Each unknown, Dk
- In this Example Find Dx , D (^) y, D (^) z as
50 4 22
3 6 2
33 9 12
−
−
− − − Dx =
8 4 22 50
3 6 2 3
21 9 12 33 − − + =
− + − =
− − = − x y z
x y z
x y z
8 50 22
3 3 2
21 33 12
−
− −
− − Dy =
8 4 50
3 6 3
21 9 33
− −
−
− Dz =
Cramer’s Method – Illustrated-
- Once We’ve Calculated all these Determinants, The Rest is Easy
These Eqns Ilustrate the most UseFul Feature of Cramer’s Method
c
x D
D
x =
c
y D
D y =
c
z D
D
z =
Dc appears in all
THREE
Denominators
Cramer’s Method – Illustrated-
- Calc the Determinants
- First Recall The SIGN pattern for Determinants
Dc = 21 ( ) (^1) −^64 − 222 − 9 ( − 1 ) (^) −− 83 − 222 − 12 ( ) (^1) −− 83 −^64
Find Dc
1146
2604 738 720
= − −
c
c D
D
[ ( ) (( ) ( ))] [( ) (( ) ( ))] 12 [( 3 ) ( 4 ) (( 8 ) 6 )]
9 3 * 22 8 * 2
21 6 * 22 2 * 4
= − − − − −
= + − − − −
Dc = − − −
Dc is LARGE → WELL Conditioned System
Cramer’s Method – Illustrated-
- Solve using MATLAB’s det Function >> EqnSys = [21 -9 -12 -33; -3 6 -2 3; -8 -4 22 50];
>> Dc = det(EqnSys(:,1:3)) Dc = 1146
>> Dx = det([EqnSys(:,4),EqnSys(:,2:3)])
Dx = 1146
>> Dy = det([EqnSys(:,1),EqnSys(:,4), EqnSys(:,3)]) Dy = 2292
All Row Elements of Cols 1-
All Row Elements of Cols 4, 2-
All Row Elements of Cols 1, 4, 3
Cramer vs Homogenous: Ax =^ b =^0
In general, for a set of
HOMOGENEOUS linear algebraic
equations that contains the same
number of equations as unknowns
- a nonzero solution exists only if the set is singular; that is, if Cramer’s determinant is zero
- furthermore, the solution is not unique.
- If Cramer’s determinant is not zero, the homogeneous set has a zero solution; that is, all the unknowns are zero
Cramer’s Rule Summary
- Cramer’s determinant gives some insight into
ill-conditioned problems, which are close to
being singular.
close to zero indicates an
ill-conditioned problem.
UnderDetermined Example-
system is the equation x +^3 y =^6
All we can do is solve for one of the
unknowns in terms of the other; for
example, x = 6 – 3y OR y = −x/3 + 2
- An INFINITE number of (x,y) solutions satisfy this equation
UnderDetermined Example-
- When there are more Unknowns than
Equations, the LEFT-DIVISION method will
give a solution with some of the unknowns
set equal to ZERO
which
corresponds to
>>A = [1, 3]; b = 6; >>solution = A\b solution = 0 2
3 6 [ 1 3 ] (^) = 6
y
x x y
Minimum Norm Solution
- When det( A ) = 0, We can use the
PSEUDOINVERSE method to find ONE
Solution, x , such that the Euclidean (or
Pythagorean) Length of x is MINIMIZED
2 2 3
2 2
2 x = min x 1 + x + x + xn
MATLAB will return the MINIMUM
NORM SOLUTION →
x = pinv(A)*b
Existence and Uniqueness
- The set Ax = b with m equations and n
unknowns has solutions if and only if
rank [ A ] = rank[ Ab ] ( ) 1
Rank[ A ] is the maximum number of
LINEARLY INDEPENDENT rows of A
- Linear Independence → No Row of A is a SCALAR multiple of ANY OTHER Combinations of Rows
Existence and Uniqueness
- Recall Rank for m-Eqns & n-Unknwns
rank [ A ] = rank[ Ab ] ( ) 1
Now Let r = rank[ A ]
- If condition (1) is satisfied and if r = n , then the solution is unique
- If condition (1) is satisfied but r < n , an infinite number of solutions exists and - r unknown variables can be expressed as linear combinations of the other n−r unknown variables, whose values are ARBITRARY
Homogenous Case
- The homogeneous set Ax = 0 is a special case
in which b = 0
- For this case rank[ A ] = rank[ Ab ] always, and
thus the system in all cases has the trivial
solution x = 0
- A nonzero solution, in which at least one
unknown is nonzero, exists if and only if
rank[ A ] < n (n ≡ No. Unknowns)
- If m < n, the homogeneous set always has a
nonzero solution
