Summary Mathematics214-Linear Algebra

, SYSTEMS RECAP Row Echelon form
② 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