OCE301 Part II: Linear Algebra lecture 2

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OCE301 Part II Linear Algebralecture 2
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Simultaneous Equations in Matrix Form
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Linear Systems of Equations
coefficient matrix and augmented matrix
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Gauss Elimination

• The Gauss elimination is a standard method for
  solving linear systems (a systematic elimination
  process).

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Example of Gauss Elimination
Textbook example pp.324-326
augmented matrix
equations
Linear system is completely defined by its
augmented matrix
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First Step
elimination of x1
add 1 times the pivot equation to the second
equation add 20 times the pivot equation to the
fourth equation
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Row Operations
Operations for equations
Row operations for matrices
Interchange of two rows
Interchange of two equations
Addition of a constant multiple of one row to
another row
Addition of a constant multiple of one equation
to another equation
Multiplication of an equation by a nonzero
constant c
Multiplication of a row by a nonzero constant c
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Exercise of Row Operations
augmented matrix
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Second Step
elimination of x2
move the second equation at the end
add 3 times the new pivot equation to the
third equation
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Third Step
upper triangular matrix
back substitution determining x3, x2, x1
work backward from the last to the first equation
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Echelon Form

• The form of the system and of the matrix in the
  last step of the Gauss elimination is call the
  echelon form.
• From the echelon form, one can conclude three
  possible cases
• No solution
• Precisely one solution
• Infinitely many solutions

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Matlab Gauss Elimination (left matrix divide)
\ Backslash or left matrix divide. A\B is
the matrix division of A into B, which is roughly
the same as inv(A)B , except it is computed
in a different way.
If A is an n-by-n matrix and B is a column
vector with n components, or a matrix with
several such columns, then X A\B is the
solution to the equation AX B computed by
Gauss elimination.
C mldivide(A,B) is called for the syntax 'A \ B'
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Matlab Example Gauss Elimination
A 1 -1 1
-1 1 -1 0
10 25 20 10
0 b 0 0
90 80 x 2
4 2
format rat A1 -1 1 -1 1 -1 0 10 25
20 10 0 b0 0 90 80' x A\b
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Row-equivalent Systems
A linear system S1 row-equivalent to a linear
system S2 if S1 can be obtained from S2 by
elementary row operations.
S1
S2
Row-equivalent linear systems have the same set
of solutions.
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Matrix a set of Column Vectors
coefficient matrix a set of column vectors
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Linear Dependence and Independence
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Linear Combination
If ck is nonzero
If one vector in a set of vectors can be
expressed as a linear combination of the others,
the vectors are linearly dependent.
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Example Problems
Answer independent
Answer dependent
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Just Determined Linear System
Just-determined system same equations and
unknowns
Over-determined system more equations than
unknowns
Under-determined system fewer equations than
unknowns
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Rank of a Matrix
The rank of a matrix A, written as r(A), is the
maximum number of linearly independent row
vectors of a matrix A.
Answer r(A) 2
Matlab function rank(A)
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Rank in terms of Column Vectors
The rank of a matrix A, written as r(A), is the
maximum number of linearly independent row
vectors of a matrix A.
The rank of a matrix A equals the maximum
number of linearly independent column vector of
A.
A and its transpose AT have the same rank.
(proof omitted)
Row-equivalent matrices have the same rank.
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Linear dependence/independence
p vectors are linearly independent if the matrix
with the p vectors has rank p they are linearly
dependent if that rank is less than p.
Applying Matlab, one uses rank to determine
whether a number of vectors are linearly
independent.
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Solutions of linear systemsExistence and
Uniqueness
coefficient matrix
augmented matrix
(one more column vector)
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Uniqueness
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Consistent and Homogeneous
A set of simultaneous, linear algebraic equations
is termed consistent if there is a solution and
inconsistent if there is no solution.
If b is not null, the set of equations is said
to be non-homogeneous, and if b is null, the set
of equations is said to be homogeneous.
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Vector Space and Dimension
A vector space is a set V of vectors such that
with any two vectors a and b in V all their
linear combinations aa bb (a, b any real
numbers) are elements of V.
The maximum number of linearly independent
vectors in V is called the dimension of V.
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Homogeneous Linear System
always has the trivial solution x 0
nontrivial solutions exist if and only if r(A)
lt n
if r(A) r lt n, nontrivial and trivial
solutions form a dimension of n r, called the
solution space.
This solution space is also called the null space
of A because Ax 0 for every x in the solution
space. Its dimension is called the nullity of A.