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SYSTEMS OF
LINEAR EQUATIONS
 A linear equation
 A system of linear equations (or a linear system) is
a collection of one or more linear equations involving
the same variables — say, x1,…., xn.
 A solution of the system is a list (s1, s2,…, sn) of
numbers that makes each equation a true statement
when the values s1,…, sn are substituted for x1,…, xn,
respectively.
 The set of all possible solutions is called the solution
set of the linear system.
Slide 1.1- 1
SYSTEMS OF
LINEAR EQUATIONS

A system of linear equations has
1. no solution, or
2. exactly one solution, or
3. infinitely many solutions.
 Matrix notation
1. coefficient matrix
2. augmented matrix
 Solving systems of linear equations
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SYSTEMS OF
LINEAR EQUATIONS

Elementary row operations include the following:
1. (Replacement) Replace one row by the sum of
itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a
nonzero constant.
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Row Reduction and Echelon Forms

A rectangular matrix is in echelon form (or row
echelon form) if it has the following three
properties:
1. All nonzero rows are above any rows of all
zeros.
2. Each leading entry of a row is in a column to
the right of the leading entry of the row above
it.
3. All entries in a column below a leading entry
are zeros.
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Row Reduction and Echelon Forms
 If a matrix in echelon form satisfies the following
additional conditions, then it is in reduced echelon
form (or reduced row echelon form):
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its
column.
• A pivot position in a matrix A is a location in
A that corresponds to a leading 1 in the
reduced echelon form of A. A pivot column is
a column of A that contains a pivot position.
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Row Reduction and Echelon Forms
 The row reduction algorithm
 Solutions of linear systems
 Existence and uniqueness theorem
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VECTOR EQUATIONS
 A matrix with only one column is called a column
vector, or simply a vector.
2
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Slide 1.3- 7
VECTOR EQUATIONS


The vector whose entries are all zero is called the
zero vector and is denoted by 0.
n
For all u, v, w in
and all scalars c and d:
(i) u  v  v  u
(ii) (u  v)  w  u  (v  w)
(iii) u  0  0  u  u
(iv) u  (  u)   u  u  0,
where u denotes (1)u
(v) c(u  v)  cu  cv
(vi) (c  d )u  cu  du
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Slide 1.3- 8
VECTOR EQUATIONS
(vii) c(du)=(cd)(u)
(viii) 1u  u
n
 Given vectors v1, v2, ..., vp in
and given scalars c1,
c2, ..., cp, the vector y defined by
y  c1v1  ...  c p v p
is called a linear combination of v1, …, vp with
weights c1, …, cp.
© 2012 Pearson Education, Inc.
Slide 1.3- 9
VECTOR EQUATIONS
 A vector equation
x1a1  x2a 2  ...  xna n  b
has the same solution set as the linear system whose
augmented matrix is
----(5)
a b.
a a
1
2
n
 In particular, b can be generated by a linear
combination of a1, …, an if and only if there exists a
solution to the linear system corresponding to the
matrix (5).
© 2012 Pearson Education, Inc.
Slide 1.3- 10
LINEAR COMBINATIONS
 Span: If v1, …, vp are in
, then the set of all linear
combinations of v1, …, vp is denoted by Span {v1, …,
n
vp} and is called the subset of
spanned (or
generated) by v1, …, vp. That is, Span {v1, ..., vp} is
the collection of all vectors that can be written in the
form
n
c1v1  c2 v2  ...  c p v p
with c1, …, cp scalars.
 Geometric description
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Slide 1.3- 11
MATRIX EQUATION Ax  b
 Definition: If A is an m  n matrix, with columns a1,
n
…, an, and if x is in
, then the product of A and x,
denoted by Ax, is the linear combination of the
columns of A using the corresponding entries in x
as weights; that is,
.
 Ax is defined only if the number of columns of A
equals the number of entries in x.
© 2012 Pearson Education, Inc.
Slide 1.4- 12
MATRIX EQUATION Ax  b
 Theorem 3: If A is an m  n matrix, with columns
m
a1, …, an, and if b is in
, then the matrix equation
Ax  b
has the same solution set as the vector equation
x1a1  x2a 2  ...  xn an  b ,
which, in turn, has the same solution set as the system
of linear equations whose augmented matrix is
a n b.
 a1 a 2
© 2012 Pearson Education, Inc.
Slide 1.4- 13
EXISTENCE OF SOLUTIONS
 The equation Ax  b has a solution if and only if b

is a linear combination of the columns of A.
Theorem 4: Let A be an m  n matrix. Then the
following statements are logically equivalent. That
is, for a particular A, either they are all true
statements or they are all false.
m
a. For each b in
, the equation Ax  b has a
solution.
b. Each b in m is a linear combination of the
columns of A.
m
c. The columns of A span .
d. A has a pivot position in every row.
© 2012 Pearson Education, Inc.
Slide 1.4- 14
SOLUTION SETS OF LINEAR SYSTEMS
 A system of linear equations is said to be
homogeneous if it can be written in the form Ax  0 ,
where A is an m  n matrix and 0 is the zero vector in
m
.
 Nonhomogeneous systems
Ax  b
© 2012 Pearson Education, Inc.
Slide 1.5- 15
SOLUTION SETS OF LINEAR SYSTEMS
 When a nonhomogeneous linear system has many
solutions, the general solution can be written in
parametric vector form as one vector plus an arbitrary
linear combination of vectors that satisfy the
corresponding homogeneous system.
© 2012 Pearson Education, Inc.
Slide 1.5- 16
SOLUTION SETS OF LINEAR SYSTEMS
 Theorem 6: Suppose the equation Ax  b is
consistent for some given b, and let p be a solution.
Then the solution set of Ax  b is the set of all vectors
of the form w  p  vh , where vh is any solution of the
homogeneous equation Ax  0.
 This theorem says that if Ax  b has a solution, then
the solution set is obtained by translating the solution
set of Ax  0, using any particular solution p of
Ax  b for the translation.
© 2012 Pearson Education, Inc.
Slide 1.5- 17
1.7 LINEAR INDEPENDENCE
 Definition: An indexed set of vectors {v1, …, vp} in
n
is said to be linearly independent if the vector
equation
x v  x v  ...  x v  0
1 1
2
2
p
p
has only the trivial solution. The set {v1, …, vp} is
said to be linearly dependent if there exist weights
c1, …, cp, not all zero, such that
c1v1  c2 v2  ...  c p v p  0 ----(1)
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LINEAR INDEPENDENCE
 The columns of matrix A are linearly independent if and
only if the equation Ax  0 has only the trivial solution.
 A set containing only one vector – say, v – is linearly
independent if and only if v is not the zero vector.
 A set of two vectors {v1, v2} is linearly dependent if at
least one of the vectors is a multiple of the other.
 The set is linearly independent if and only if neither of
the vectors is a multiple of the other.
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LINEAR INDEPENDENCE
 Theorem 7: Characterization of Linearly Dependent
Sets
 An indexed set S  {v1 ,..., v p } of two or more
vectors is linearly dependent if and only if at least one
of the vectors in S is a linear combination of the
others.
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SETS OF TWO OR MORE VECTORS
 Theorem 8: If a set contains more vectors than there
are entries in each vector, then the set is linearly
n
dependent. That is, any set {v1, …, vp} in
is
linearly dependent if p  n .
 Theorem 9: If a set S  {v1 ,..., v p } contains the
zero vector, then the set is linearly dependent.
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1.8 INTRODUCTION TO LINEAR
TRANSFORMATIONS
 A transformation (or function or mapping) T from »
to » m is a rule that assigns to each vector x in » n a
vector T (x) in » m.
 A transformation (or mapping) T is linear if:
i. T (u  v)  T (u)  T (v) for all u, v in the
domain of T;
ii. T (cu)  cT (u) for all scalars c and all u in the
domain of T.
Slide 1.8- 22
n
LINEAR TRANSFORMATIONS
 Linear transformations preserve the operations of
vector addition and scalar multiplication.
 These two properties lead to the following useful
facts.
 If T is a linear transformation, then
T (0)  0
----(3)
T (cu  dv)  cT (u)  dT (v)
T (c1v1  ...  c p v p )  c1T (v1 )  ...  c pT (v p )
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1.9 The Matrix of A Linear Transformation
 Let T : » n -- » m be a linear transformation. Then
there exists a unique matrix A such that
T (x)=Ax.
In fact, A=[T(e1), T(e2), ….., T(en)], where {e1, e2, …..,
en} is the standard basis for
»n .
The matrix A is called the standard matrix for the linear
transformation T.
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The Matrix of A Linear Transformation
 A mapping T: Rn-Rm is said to be onto if each b in
Rm is the image of at least one x in Rn.
 A mapping T: Rn-Rm is said to be one-to-one if
each b in Rm is the image of at most one x in Rn.
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The Matrix of A Linear Transformation
 Let T : Rn -- Rm be a linear transformation. Then T
is 1-1 if and only if the equation T(x)=0 has only the
trivial solution.
 Let T : Rn -- Rm be a linear transformation and let
A be the standard matrix for T.
1) T is onto if and only if the columns of A span Rm.
2) T is 1-1if and only if the columns of A are linearly
independent.
Slide 1.8- 26