Class notes Linear Algebra

Lincer Combinations of Vanables
Theorem
1 5 Gauss' Method
.
:



C X,6
,
CXcb . . . Xn c = R ↓ liner system can be
changed to another by one



of these operations
A liner equation in neveriables has the form 1) Our equation swapped with another
) One equation
2
multiplied byis a non-zero scaler

2X
,0 Xb ...


62Xn = d 23d = IR 3) An equation is replaced with the sum of itself is
, ,



a sealer multiple of another equation


* lineer system in neveriables has the form : &
inition These operations or called now operations ,




elementaryrow operations ,
or Gaussian Operations
.




a X +9 24 ... +a
nXn = d
, , ,
, , ,




In eachvor of alimestem
2 X 6 az , 2X26 +az nXn =
da the first ona
,
...

c ,, ,
,




&
k ,, X ,
+
9 2
X
<b
...

+ak nXn = dk
, ,




* lineer system is
in echelon form if the /
We say an notuple (S
,,
Sc
,
"Sn) is a solution the in each now is to the right of the leading verable


previous liner system if it solves each equation . in the preceding equation

W


. See
Definition Al"real") matrix Goal
: is a kxn array of
real :
Augmented matrix into

numbers (K rors
,
n columns) echelon form using now




operations . Due in


I matrix be echelon form
agmented in a natur , you can


the do algebraic substitutions
column represents constants in a
line system

and whos first in columns represent the to solve for all

offerent in an n-variable linear system variables
, provided that
there is a solution




example
kshut2)
↳ A : Geekho , man
o e




nod = 10


Sn -d = So




(sidl) It's 1) .
&
...
>
-

,
,