Class notes for Math 314 Linear Algebra

Systems of equations
Y = MX +6 - line where two lines meet is the
9 + by = C -
line solution her boon equations.


if two ineslequations don't meet,
There is no solution

If bein likes are the same , where

are infinit soitions


equations in one form dx + dexz = 6 are linear.

no variables on one side

a list of all possible solutionsbo a linear system
Is called a
solution set

linear systems can
have no salvim ,
one solutions, a

infecte solutions (consistent or inconsistent)


(74x [J 2x4




eX .
X, -2xz + xz = 0 solve .




2x2 -
8x3 = 8 = 0x1 + 2x2 -
8x3 = 8

Sx1 - 5x3 = 10

, Coefficient Matrix Avamented Matrix




I· [ I
2 21

o
O
-
-




2

S0-510

X, +2 +
3
X +
2x3 =




replace equation 3 Win
-

S(291) + 293
10
=
-
Sx,+ 0x2 = 0 + Sx-St3 =




I I
1
-
2 10
293 : 1642-1043 = 10
- 8S
02
-




10-10 10
replace equation 2 won legt g


2x2-81 :8 -X-4X3 : Y


replace earation with-10leaztess
(a)
3


30X32-30 ,
X3 = -1

(xz -

4xy = 4 Yz + y =4




↳
Xz = 0




·
x
z(0) + )( ) = 0
·




,
-




X -
1 =0
,

X, =

, ElementaryNew operations

1 . replacement [replace the lowI by the Sum of itself
and of anainer
a multiple

2.
Interchange (Swa two rars)

. scaling
3 (Muriply a new by a mustifical


Two marries are few equivalent If operations
exist to transferm them from the to The sincer.

(A -
B)

is the system consistent ? (does It have a sartial
If consistent, is there I or infinite sontios ?

ex . Determine If the following system is consistent:
Xz 4xz = 8



[ii]
-




x
,
-
3xz + 2x3 = /
↑ x1 -
8x2 + 12x3 = /


Interchang row 1 and 2 to remove the zero from rowl




]
32

[ I
1 -



Chang raw 3 With row 3-yearI
01 4 g
= N Car's
-




remove the 4 In
4 -
S 12 (
T

,=
& S-e R3-wide to remove the
U In R S




Y 3x2 + 2x3 =




[1 )
-

/




:
Since we can
xz -
4x3= S
= Now selve
:
,
-




W
12 = IS Consistent
+3 system


armented has beentransformed ,
2X .
This Matrix
IS IT consistent ?
/



[, 7
2 -
32 2x
-
3xz + 2x1 =



Ixi-Uxz = S can save


0x 3 = 1S
:




Non Zero row : Any rew with at least 1 ren-Icro value
leading entry : Lefimosi non-zero value In non-zero No
Now ecuelon A Matrix
term :
is In warecocion form If
1 . any nen-zero low is above Zero raws.

2
. each leading
entry appearsIn a en to the

right of leading entry aboveli .




.
3 below a leading any
All entries in a column
are all zeros




[
O=
leadhto
J
-x X x x

000XX
000X
· o o x Ref