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Systems of Linear Equations #2: Gaussian Elimination
Outline: Gaussian Elimination (Ax=b → Ux=c)
1) Review mechanics
2) Review modes of failures
3) The general pattern of GE
3) GE using Matrices (E and P)
4) General Rules of Matrix Matrix Operations
Addition: (A+B)
Matrix Multiplication : AB
Matrix Powers : A^{P}
Gaussian Elimination on a 3x3 system of equations :
Example ( with row exchange):
x + 2y + 4z = 1
2x + 4y + 2z = 2
6x + 10y  z = 8
Matrix Vector form:
Augmented Matrix form:
Failure of Gaussian Elimination (review):
Example 2:
Example 1:
The overall Pattern of Gaussian Elimination:
A recursive Algorithm that reduces Ax=b to Ux=c
If successful: U will contain n pivots on the diagonal
( unique solution )
If Fails: U will contain at least 1 zero on the diagonal (no or ∞ solutions)
Gaussian Elimination using matrices:
Two Basic operations:
1) Elimination steps to zero out a_{ij}
2) Row exchanges to repair tempiojrary failure
BIG IDEA! We can design Matrices (E and P) to do this for
us
Elementary Elimination Matrices E_{ij} :
MatrixMatrix Multiplication: C=AB
Column View:
Row View:
Other Examples: Diagonal Matrices
Left Multiplication: DA
Right Multiplication: AD
Does AD=DA (is matrix multiplication commutative ?)
Gaussian Elimination:
a sequence of Elementary Elimination Matrices
Point: Now we're doing real Linear Algebra!
( Algebra of matrices and vectors, not arithmetic )
Permutation Matrices
Again: AB≠BA in general
Gaussian Elimination:
a sequence of Elementary Elimination Matrices
Point: Matrices Do things!
An important Digression:
General rules of Matrix  Matrix Operations
Matrix Shape:
Matrix Matrix Addition : A+B
Properties of Matrix Addition: (follow from scalar and vector addition)
Matrix Matrix Multiplication: C=AB
Examples ( with numbers ):
Properties of Matrix Multiplication:
Theorem: Matrix Mult is associative.
If A,B,C are matrices of appropriate shapes, then A(BC)=(AB)C
Proof (sketch): Show A(Bc)=(AB)c
Another important Digression:
Operation costs of Matrixvector and MatrixMatrix multiplication
( order matters !)
An important Digression:
General rules of MatrixMatrix Operations
Matrix Powers: A^{P}
The matrix Inverse: A^{1}
Summary :
Gaussian Elimination using Matrices
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