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System of^ Linear^ Equations
G,x1 + 92x2 + ...andn =^ b^ +^ a^ =^ real^ numbers
+ x = variables
2 x (^2) system 2x (^3) system 3 x (^2) system
- 2(X1 + 2x2 = 5) x1 - x2 +
x3 =^2 X^1
+ xz = 2
#^ xz^
xe^2
- x3 = 4 X 1 - xx = 1
- 2x1 - 4x2 =^ - (^10 3) x -x2=^8 1 Si = 4
- x 2 = - (^2)
with (^) X1 = 4, impossible x to^ satisfy both equations
(1,2) intersection^
no intersection:inconsistantsystem
X, +^ x^ = 2 I
x +X2 = 2
I
x1 -^ x2^ =^2 x
+ xz = 1
12,0) 5 F^0 -parallel lines X1 + xz =^2 I --
- X1 - x2 = - 2 1 -^ 0,0)^ Every Linear^ System of^ equations + (^) same (^) line can only have:zero,^ one,^ or infinitly (^) many solutions 5x, + 3x2 =^ -^12 513) + 3x= =^ - 12 3( - - x, - x (^) =^ =^ b)^ 3x2 = - 12 - 15
5x, +^ 3x2^ =^ -^12 A 91
-^2 = 18
2x,
1 5 = 313, - 9)
x + y+^ 2z^ =^9 x - y + z =^2 2x + y -^37 =^ -^5 per (2x + 3z =^ 11(^ 2(1) +^ 3z^ =^11 (1)^ + y +^ 2(3)^ =^9 2
3z =^9 y =^9 -^7
S = 2(1,2,3)
i
be parallel ise
Augmented Matrix x1 + 2x2 + x3 =^3 E I^ I (^) s I
3x1 - xz - 3x3 =^ -^1 - -^ I
2x1 + 3x2 + x3 =^4 Row 3 x^3 E
Elementary Row^ ofoperations
· Add/subtract any two^ rows^ together
Multiply (^) any row^ by KF ·Row (^) Swap, (^) exchange any two (^) rows Echelon form x + y - z =^7 I^ ->^ needs to^ be (^) zeros, kill x - y + 2z = 3 -^ I^ bellow (^) using pivot (the^ ones 2x + y + z = 9 I (^) : 2 R, - Rz - > (^) Rz I
- 2R, + R3 -> (^) R FO I i) !-
- I (^) ->
- 2(1 + 2 = 0 ' (^) -5=z - 2(1) + 1 =^ -^1 -1-2 =^ -^3 - 2(- 1) +^1 =^3 I I I^ -^ I I 0 2 is^ 0 3 - 1 3 een^ () 0 +^ 2(0)^ =^0
2 + 2(- 1) = 0 x + y - z = 7 = 2
- 3 + 2(3) = 3 2y - 3z = 4 i
I is)
- needs tose zeros I = I I I -))me I^ 3 =(i) ⑧ 2 ⑧ re ⑧ 0 ⑧^0 3(1) -^0 =^3 0 + 0 =^0 3(1) -^2 =^1 2 +^0 =^2 (^3) - 1) - (- 3) = 0 - 3 +^3 =^8
3 I I 6 ⑧ 8 2 2R,^ -R2^ +R^ I^8 I (^) ⑧ 0 of-I 2(3) -^0 = 6 0(1) -^0 =^0 l 2(1) - 2 = 1 ·Reduced Row^ Echelon form ⑧ (^) -> (^) needs tobe ones I (^) : (^1) -^2 · 2 I in. 6 ⑧ I
:/ 8 8 I (^0 ) (5) =^1 12 =^1
InconsistantSystem: I (^) =5) ie, (ii) (^) E ConsistentSystems: infintiymne,'' (^) "=a* ee I (^13) parameter x, + Xz =^1 x, = 1 - a
a
reque, (^) polition, -x x, =^1 -^1 x, =^0 Xz =^1
Undetermined System
-> Usually has^ infinitly^ many solutions,^ due^ to^ free^ variables =>
Typically consistantsystem
typically shaped like^ such I (^17) I ⑧^ -^5 I^ x (^) I^ - 5x5 = 1 x 3 = free I (^) xz + x3 = 4 I·.:^ t (^0) = 0 LetX3 = 2 Letx3=^ - 6 x, =^1 + 5(2) x, =^1 +^ 5(- 6) -z I
In 2 x^2
= 18
x, - 2xz -^ x3^ + 3x4 =^0 [ 2x, + 4x2 + 5x3 - 5x4 =^3 I 3
- Ry 3x, - 6x2 - 6x2 + 8x4 =^2 I 0 + 5 I R2 + Rz =Rz I
:(E)
+ inconsistant
Iiii)
3R, - Rz - > R
Ii?."13]
- I⑧:() -R3 -Rs -- (08:(5] X, +^ 3x^ + 4x3 =^7 ↑ x, +^ 3xz^ +^ 4(3)^ =^7
x3 =^3 x, + 3x3 =^ -^5
no (^) pivot, so free x (^) z^ =free^ x, =^ -^5 -3x s = 2 -5- 3x,a,3/a[R
x, +^450 = xz +^610 x, - xz = 160 xz +^520 = x3 +^480 I
xz - x3 =^ -^40
⑦ (^) x3 + 390 = xn = (^600) x3 - xn = 210 x, +^640 =^ x,^ +^310 Xu - x, =^ -^330
⑧ I
I
I - 100
I
R, -Ru - Ru
W 0 I
I - 16o (^) - I 160 I ⑧ -1. I
- 48 Rz + R -^ >Rn^ ...^ R^ - Rn^ + R .I
I
I
W 0 I W 0 I -
2 i
- I 0 - 178 W (^) ⑧ - I - 210
⑧ (^) -1. 10 -^ I I ⑧^ I^ 1
0 2 C
·iii). (^) inta,I
W 0 I
I xy = G
X, =^330 +^ x
(^1) ·(i) s = 2330 + 2,170 + x,210 + x.a(ae
Overdetermined system
-> (^) has (^) more (^) equations than (^) amountof (^) variables
typically inconsistant/no^ intersection(s)
typically shaped^ as^ such ( (^1) =(2:1)e(!:] em x, - xx =^3 I E (^) :) (: "yes,
I
. I I S = 0 0 F^3 inconsistant
Algebraic functions Two (^) matricies Aand B (^) are (^) equal off (^) they are ofthe same size (man):aij=bij · Scalar (^) Multiplication A = L
I
=n =
(55) 3A = 6810 lii] · Matrix Addition
aij = bij=(aij^ [bij]
(ii)
(-2) -> (^) undefined (2=)
(is)
(isis)
(i) · Matrix (^) Multiplication
- (^) Row Column (^) way -Six.(Yx=)"(axo
-^ B Fii).. ( A (in (^) Jame BA FAB
a =(3.^ -2) + (1.2) +^ (3.1)^ =^ -
9.2 =^ (-^ 2.^ - 2) + (1.4) +^ (3.-^ 3)^ =^ -^9 92 =^ (3.4)^ +^ (2.1)^
- (1.6) = 20 an = (-^ 2.4)^ + (4.18 +^ (-^ 3.6)^ =^ -^22
↑ - scaller wway. (*()-e I
- (iii) s B I i AB = [ ]axz am = 2(3,2) + 1(2,4) + 3(1,- 3)
amz =^ 4(3,-2)^ +^ 1)2,4)^ +^ 6(1,^
= 20, -
- Column^ Way
itcolumn
111 k.) :)^ I (^) a
- (iii) s B I (^) A (in (^) Jame an = 3(ii)
2(i) " () -(ii)
an =
3(3) -(2)
EV, (^) V2, [2, V53, EVs, (^) Vn3, EV3, (^) V53,[Vn, V I 0 W^2 Vi A =^ I 0 W^ D^ I^ Vz ② *^ I^ I (^) VS I
⑧ 0 I^2 I
I V- Adjacency Matrix^ B (^) I (^) I 10 V V. Vz^ Va^ V^ V
At -^0 I^00
10 ↓ ⑧^ I AT = A= symmetrical I
O 0 I I
D (^) I
O I^ ↓ I
o I^ I^0
Vectors: -> (^) Quantaties thathave magnitude and^ direction
- magnitude:distance,^ mass,^ volume,^ and^ temperature
-> i.e:
volacity, force
-> (^) vectors are (^) exactly one row /column · = (i) =(21]
horizontal rector
Column rector
: Vector Addition = - (1) = (i) +r^ = (2 in)
⑧ I - · +E
I -(i) Vector (^) Multiplication *= 2
(i)
Linear Combinations
-> (^) x=ar + Xzart... +^ Xzan, X: tR,: ER"=linear combination
-> (^) Spanda....,an3= (^) all linear combinations of (^) ai,...,an ERM · (i):9(). It. (^) Ill x, -3x2^ + 2x3 =^ -^1 2x,
- 5x2 - x3 =^4
- 4x, + 13x2 - 12x3 = 1) I - 32 his (^) in I lies (it mee Z I - (^3) x 3 = 1 :e I x (^) z =^11 (^1) I I^ :5)^ x,
= 30 A linear (^) combination ·(i)
3x, + 5x2 =^0 +Rx - > Rz
2x, +^ X2^ =^2
Elizen. (21:)= 4x, - 3x2 =^0 ii) in a re I
Linear Combinations^ Equation:Ax=^5 A xY^5 [a,, am, ..., (^) a,n] all [aix, (^) and.^..^..^ anXn]^ exn Ax = (^5) in vector (^) form atten I
I
= I
D
I x, =^ - 2a Y = I ....: -(y)-i)- 0 t-)
C
Sto Vector (^) Multiplication
A
I is^ i ()
- (i) + 1(2) = -(i) =(i) =^5 Have infinitly many solutions in
