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More On Linear Systems
EM 120
Lecture 31
1EM120 Lecture 31 Dr C K Tan
Overview
We consider a special type of linear system,
the Homogeneous linear system, where the
right hand sides are equal to zero.
Well also look at the Gauss Jordan method for
solving systems of equations and finding
inverses.
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Motivation:
Homogeneous linear systems occur frequently
in practical engineering problems and also
give rise to the notion of the null space of a
matrix which we will study later.
The Gauss Jordan procedure is used to avoid
the need for back substitution and can also be
used to determine the inverse of a matrix.
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Key concepts
Homogenous systems.
Gauss Jordan method.
Elementary matrices.
Computation of inverses.
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Contents:
The solution of homogeneous systems.
The Gauss Jordan method to solve linear
systems.
Elementary matrices and their relation to
e.r.o.s.
Using the Gauss Jordan method to determine
inverses.
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Outcomes:
You should recognize the special nature of
homogeneous systems and understand the
associated terminology.
Ability to determine the inverse of a matrix by
the Gauss Jordan method.
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Homogeneous Systems Homogeneous systems are simply linear
systems where the right hand side of each of the equations is equal to zero
Always consistent, r([A|b]) = r([A|0]) = r(A), regardless of what the matrix A is. Trivial solution: x = 0
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Or Ax = 0
Possible solution sets
(i) If r(A) = n, the trivial solution is the only
solution.
(ii) If r(A) < n, there are infinitely many
solutions (trivial and non-trivial).
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Example
Homogeneous System:
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Solution Set
As r(A) = 2 is less than the number of
variables, n = 3, there are infinitely many
solutions.
Let
Then
And
Solution set: which includes
the trivial solution
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Reduced Row Echelon Form
Gauss Jordan method, the augmented matrix
of a system into the reduced row echelon
form. This simply means that all leading
entries in the matrix are 1, any leading entry
occurs to the right of the leading entry in the
row above, all zero rows are at the bottom of
the matrix and any column containing a
leading entry has only zeros in the remaining
entries.
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Elementary Matrices
An nn matrix is called elementary if it can be
obtained from the nn identity matrix by a single
elementary row operation. e.g. E1, E2 and E3 below
are all elementary matrices:
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Effect of elementary matrices
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Same Effect as applying E R O s
Observation: Let E be obtained from Im by a particular e.r.o. Let A be any m n matrix. Then performing the same e.r.o. on A will result in the matrix EA. (i.e. an e.r.o. is equivalent to a matrix multiplication by the corresponding elementary matrix).
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Inverse of an elementary matrix
every elementary matrix is invertible and its
inverse is also an elementary matrix (and
simply corresponds to the e.r.o. which does
the reverse of the original)
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Calculating Inverse Matrix by Using
Gauss-Jordan Method
Matrix A (nxn) must be invertible or non-
singular.
Start with the augmented matrix [A | I], apply
E R O s until the left hand side becomes the
identity matrix. Then the right hand side is the
inverse of matrix A.
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Singular Matrix: No Inverse
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