I
Linear Combinations of Vanables
C X,6
,
CXcb . . . CnXn C= I In each row of a liver system , the first nu-zers
term is called the leading -able
A liner equation in neveriables has the form
* lineer system is
in echelon form if the /
d 23d IR
2X , 0Xb 62Xn =
in each is to the right of the leading verable
... =
, ,
now
in the preceding equation
* lineer system neveriables has the form : Goal echelon form using
in :
Augmented matrix into now
operations
.
Due echelon form substitutions
in
, you can do algebraic
a X +9 24 ... +a
, nXn = d ↓ solve for all variables
provided that there is
, , , , , ,
2
c ,,
X
,
6 az , 2X26 ...
+az ,
nXn =
da a solution
&
k ,, X ,
+
9 2
X
<b
...
+ak nXn = dk
, ,
Definition .
A set S is a collection of objects called
We say an notuple (S
,,
Sc
,
"Sn) is a solution the elements or members .
previous liner system if it solves each equation
.
The set of all real numbers is denoted
R
Definition. Al"real") matrix is a kxn array of
real ↳ S2 is a real number so can be shown
,
numbers (K rors
,
n columns) "SER"
An matrix whose last "T2-IR" translates to 52
aumented is a matrix is an
column represents the constants in a
line system an element/member of the real .
number set
and whos first in columns represent the
offerent in an n-variable linear system Set builder notation
↳ set :
you can describe all elements in a
Theorem Gauss' Method
Formula inton
.
↓ liner system can be
changed to another by one s =
E
of these operations
1) Our equation swapped with another canplya
) One equation
2
multiplied by is a non-zero scaler
3) An equation is replaced with the sum of itself is s =
3 I 3
a sealer multiple of another equation
un un
element your'e property of element
Definition These operations selected
.
or called now operations, selecting
elementaryrow operations ,
or Gaussian Operations.
,2
Examples .
S =
G1 ,
2
,
4
, 83 Add metirees of same size
,
entry by entry
32" /k 33
(i)(iii)
S =
= 0 1
,
2
, .
S =
ExtR/x is
rational]
S29S
-(12) is)
Definition .
Variables (in echelon form) that be
not a Subtract likewise
leading variable are a free viable.
Scale multiplation ,
let I be a sealer
(213)
x = 7z +
02:1 A
Icacalist
H X -
-zo4
3)70)02 = 1
2(4) :
(ii)
33367 = 1
** You can multiply 2 mutrices A B ,
if
Na
=
k or Ka =
NB
() A .
I 123)B
456
=
( ))
,
2 +3
3)(ii) 1 2 3
3x/
=
AB =
(13 1)02 (5) =
If a matrix has n rows and K columns we
,
can write :
A Suppose A
(
is mabrix B
&
nxk and kXm
alia
an is a
matrix
.
The product of As B will be an nxm matrix
AB C let hij be the entry in the ith
an
=
column
---
now
,
i
and column of C
an ,
1 an
,
2 ...
jth
between Cij dot product of the ith of A and the
add double digit indices.
=
* a commu
now
of B
jth now
,3
GausCordan EliminationRedmedchelonFoa 16 : 3) ↳ & 3 no solutions
,
if addition echelon form
,
in
to
being in
#85 )
each leading entry is
I and is
%
the only nonzwo entry in its
column 1 pirot free
.
,
y3z are variables
Definition . The
leading entry is also called a pirot ↳ Infinite solutions
Example.
Definition. A linear equation is
homogeneous if it has a
I (ii)
I 2 constant term of A lime
zero.
system is
·
J
homogenous if every exaction is homogeneous .
O
oooo
makethisleabo
X +
y + 2z = 0
I I
.
it X +
Y 27 = 0
X
-
37 = 0
im
pivot is I
,
and is the only non-zero entry & 2
,
9 0
, ... Gal e
, ... ER3
in
its column .
Where I is a
particula solution to the lime system
Non-example -
Example and where the number of rectors (B) is equal
I·
.
%]
to the # of free voiables the system has
after Gaussian Reduction
Lemme.
For any homogeneous system there exists
,
Note . vectors Bi such that the solution set
has the formi
# of free voiables
& , , ... ,
... -R3
↳ If there is at leastI free vorable in a
has Whoe K is the number of free veciables after
system with no contradictions , then the system
solutions Gaussian Reduction
infinitely many
Linear Combinations of Vanables
C X,6
,
CXcb . . . CnXn C= I In each row of a liver system , the first nu-zers
term is called the leading -able
A liner equation in neveriables has the form
* lineer system is
in echelon form if the /
d 23d IR
2X , 0Xb 62Xn =
in each is to the right of the leading verable
... =
, ,
now
in the preceding equation
* lineer system neveriables has the form : Goal echelon form using
in :
Augmented matrix into now
operations
.
Due echelon form substitutions
in
, you can do algebraic
a X +9 24 ... +a
, nXn = d ↓ solve for all variables
provided that there is
, , , , , ,
2
c ,,
X
,
6 az , 2X26 ...
+az ,
nXn =
da a solution
&
k ,, X ,
+
9 2
X
<b
...
+ak nXn = dk
, ,
Definition .
A set S is a collection of objects called
We say an notuple (S
,,
Sc
,
"Sn) is a solution the elements or members .
previous liner system if it solves each equation
.
The set of all real numbers is denoted
R
Definition. Al"real") matrix is a kxn array of
real ↳ S2 is a real number so can be shown
,
numbers (K rors
,
n columns) "SER"
An matrix whose last "T2-IR" translates to 52
aumented is a matrix is an
column represents the constants in a
line system an element/member of the real .
number set
and whos first in columns represent the
offerent in an n-variable linear system Set builder notation
↳ set :
you can describe all elements in a
Theorem Gauss' Method
Formula inton
.
↓ liner system can be
changed to another by one s =
E
of these operations
1) Our equation swapped with another canplya
) One equation
2
multiplied by is a non-zero scaler
3) An equation is replaced with the sum of itself is s =
3 I 3
a sealer multiple of another equation
un un
element your'e property of element
Definition These operations selected
.
or called now operations, selecting
elementaryrow operations ,
or Gaussian Operations.
,2
Examples .
S =
G1 ,
2
,
4
, 83 Add metirees of same size
,
entry by entry
32" /k 33
(i)(iii)
S =
= 0 1
,
2
, .
S =
ExtR/x is
rational]
S29S
-(12) is)
Definition .
Variables (in echelon form) that be
not a Subtract likewise
leading variable are a free viable.
Scale multiplation ,
let I be a sealer
(213)
x = 7z +
02:1 A
Icacalist
H X -
-zo4
3)70)02 = 1
2(4) :
(ii)
33367 = 1
** You can multiply 2 mutrices A B ,
if
Na
=
k or Ka =
NB
() A .
I 123)B
456
=
( ))
,
2 +3
3)(ii) 1 2 3
3x/
=
AB =
(13 1)02 (5) =
If a matrix has n rows and K columns we
,
can write :
A Suppose A
(
is mabrix B
&
nxk and kXm
alia
an is a
matrix
.
The product of As B will be an nxm matrix
AB C let hij be the entry in the ith
an
=
column
---
now
,
i
and column of C
an ,
1 an
,
2 ...
jth
between Cij dot product of the ith of A and the
add double digit indices.
=
* a commu
now
of B
jth now
,3
GausCordan EliminationRedmedchelonFoa 16 : 3) ↳ & 3 no solutions
,
if addition echelon form
,
in
to
being in
#85 )
each leading entry is
I and is
%
the only nonzwo entry in its
column 1 pirot free
.
,
y3z are variables
Definition . The
leading entry is also called a pirot ↳ Infinite solutions
Example.
Definition. A linear equation is
homogeneous if it has a
I (ii)
I 2 constant term of A lime
zero.
system is
·
J
homogenous if every exaction is homogeneous .
O
oooo
makethisleabo
X +
y + 2z = 0
I I
.
it X +
Y 27 = 0
X
-
37 = 0
im
pivot is I
,
and is the only non-zero entry & 2
,
9 0
, ... Gal e
, ... ER3
in
its column .
Where I is a
particula solution to the lime system
Non-example -
Example and where the number of rectors (B) is equal
I·
.
%]
to the # of free voiables the system has
after Gaussian Reduction
Lemme.
For any homogeneous system there exists
,
Note . vectors Bi such that the solution set
has the formi
# of free voiables
& , , ... ,
... -R3
↳ If there is at leastI free vorable in a
has Whoe K is the number of free veciables after
system with no contradictions , then the system
solutions Gaussian Reduction
infinitely many
