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Summary Mathematics214-Linear Algebra
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This notes contain all the work done in Mathematics 214 regarding Linear Algebra.
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, SYSTEMS RECAP Row Echelon form② G.J to getto
①Setup Augmented Matrix
③ Detect
consistency
Fx)(x, -
3x7 5x3
+
7
=
z -
3 52 R2 11.1 3 ④ Detect free variables
-P
1 1 3R,"2 352 ⑤Solve backsubstitution
3
using
+
xz xz
-
x,
+ -
=
R. zRz
-
11 -13
x,+xz -
723 3
=
0 57
-
4
5xx 7)(3
-
4
-
= -
+
set >c3 =t -> (x] is a free variable)
=- 5xz -
= 4 -
7xz =
- 4 -
7t
x =
5 t +
( 5t)
STEPS:1 Set Augmented matrix x =3 7z x3 3 +
t
-
- +
up
= +
2 Reduce to row echelon form
3 heading entries & consistency
1. Detect any free variables
5 General solution
Homogenous vs non-homogenous systems
1
It N
2 3
xy
2x + 7z
+
x, -
2xz xz
+
xy
+
0
=
x, -
1
=
2x, txz -
7z xu
+
0
=
2x, +
xz -
xz xu 3
+
=
↓ find two solutions to H.v, Ve
2. find two solutions toN: W., We
3. Compute a) v, vz
+
b) wi + wec) v +w, d) 3.Ve) 3.W,
4. Which of them is a solution to It?or N? free variables
me
H 1
:
-
2 1 | 8 1 -
2 | | O 1 -
110R +2R210- --50
⑧
05R. 1=5 50 15 0
-
11 O 01
-
21
-
R2 R2 -2R,
=
05 -
3 +
Setxz 0,xu t =
=
Use R2 1.2.
= - 5x3 5x - 0
=
Use R1 1.x.
=
5x3 5xu 0
-
+
=
(
-
E1 5t -
0
=
x- 50 5t 0 + =
x= 3r Et +
x
=
5p Et -
VI Vz
x5r 5t -
80 =
o r5
=
-
2 v5
=
1
xz 30 5t +
t 0
=
g
t 5 =
4 t0
=
3
=
o 5 5
(3 ↑
t O 5 o
xu
·
!
N: 1 2 / C
solutions.
-
Get as two
21 +
13 W, =
Wz =
I
So,
V, =
<...
v. =
! wz-
3) a) I
5
b) d
3
a)
-ds -·3 e)
↓ ↓ ↓ ↓ ↓
4) H None N H None
, LINEAR COMBINATIONS
Homogeneous linear Systems have the following special properties:
1. is always a solution
2. if v, we are solutions then VitVe is also
3. If v, is a solution, and cell then c.v, is also a solution.
:. ALWAYS consistent
aGIR
A consequence is that for any set of solutions ....... In, the linear combination a.+ackrt...tank,
is also a solution.
xx) Compute the linear combination 2(x" 2x
+ -
1) -
3(x2 -
x 2)
+
+(1x 3x 3x 6
2x 2
+
-
- -
-
x+7x -
8
!
Ex) Compute the linear combination 2.3
- I
Ex) Write cost as a linear combination of 1, cosi, since. Whatis the "vector-equivalent"?
=cos -
Sin 2COS -
1
9
=
Or
I -
I -
dependance: i
8
Linear
I
linearly dependant
I
since- COSC: is
-
1-
8 ! Y
xx) 7 linear combination of and
?
Is, a
Suppose that is: a bY :Need to determine is
system is
consistant or not
8 a 2b
+
Then
b
I
2a +
- -
a b
+
m
-
-
the
system
2 S
This equivalent arzb 8
is to I
=
128
24 + b 7
=
-
2 I 7 R2 IR3
+
O 39
-
a b1
+
=
-I I 1
-
R3 + R, 0 3 q_
Itis consistent, and thus their - -
exists a solution for a, b and thus I 28
S Re O I 3
i is a linear combination of $ R3 -
Rz O S 0
Ex Is (xc+2) a linear combination of (1+2x+ x2) and (1-x2)?
Suppose thatx+z a(1 + 2x x)
=
+
b(1 x)
+
-
=(a b) (2a)x
+
+
(a b)x2
+
-
Then ab +
=
2 I I I I 2 I I 2: System is inconsistent
24 1
=
2 O
? R2-2 RI O -2- 3 O -2- 3 This means that xc+2
8 2 I
b0 1- 1 R3 2 8 0 not a linear combination
-
a RI 0 is
R3
-
- = -
- R2
, Method
1. Setup an
equation if there were a linear combination
Turn itinto linear
2. a
system
3. Solve the linear
system
Ex) Is it a linear combination of1-2x and 2-3x?
Suppose that ( =
a(1-2xi) +b(z -
3x2) 2(1 2x)) (z 3x2)
=>
-
+
-
=(a 2b) + -
(2a 3b)x
+
= x2
+
Then a 2b 0
+
=
2a 3b
+
=
- 1
ex) ((,y,z) G(RY:x 4E =
-
y23
↳
Hyperbolic Paraboloid facing to the c-axis. -
...
1. Given v, in IR, whatlinear combinations can you form V,
U., ...
In:C,U+C2K2+...+CnUn CEIR .
X:C, X Cine)
(8])..
n 1 =
The span (setofall linear combinations ofX, is a line exceptifvi=
whatlinear for a
it
13,
combinations can you
Itwill be a plane IR. Exceptif one
in is 8 or if
one is a scaler multiple ofthe other.
Then we
geta line, or if vi =
V2 8. Then
=
itis [8}
3. Given ↓, , 1s c/R3, Whatlinear combinations can you form?
exi) 42
=
13
=
8
=
Then
58 3
exe)
v, Y, Vs . Then it's
=
v =
=
aline
ex3)1 =
0, v=
2. 8 I =
Then it's a plane
=8,x
↓
exc1, 0, =
8. =
Then it's the entire IR3
SPAN: Setofall linear combinations thatcan be formed those vectors.
using
ex) Whatis the span of (i) in 13?
span)!]) is a line. Span ((!])=3[-] ceRRE
Ex2) Whatis the span of1 and Iin IR3?
span)_",i") Ea
=
b
+
a,be(R3
This is a plane in IR
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