microsoft powerpoint - lecture 31 more on linear systems

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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.

EM120 Lecture 31 Dr C K Tan 2

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.


Key concepts

Homogenous systems.

Gauss Jordan method.

Elementary matrices.

Computation of inverses.


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.


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


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).


Example

Homogeneous System:


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


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:


Effect of elementary matrices


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).


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)



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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EM120 Lecture 31 Dr C K Tan 19 EM120 Lecture 31 Dr C K Tan 20

Singular Matrix: No Inverse



microsoft powerpoint - lecture 31 more on linear systems

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