Fo DA Linear Algebra•

Review # I :

L I° Vectors

,Matrices

,

Addition, Multiplication

Vectors & Matrices

vector N = ( ya ,vz

, . . .

,ud ) e- IRA

aScalar

n data points✓ = ( VEG

.ve

. . . ud ]

d dimension rowattributes column vector vector

matrix nxd matrix A EIR'd

N rows a , care ,. . . an C- Rfd

* eaiai . - is

vectorv =

fizzy EIR"

mat a ,=(3 ,

-7,

2) HR"

A- - C ?-

I⇒ e IR" '

transpose i change roll if rows

& columns

At --f÷÷s]eR"

a6.5 ,7.5 )

i. 5- 112?

Xz byA

"

""

"

as :* .⇒

-÷. !#¥:S.

A- Ea.

'

ER"

Lineations3 x

,

- 7×2 t 2×3 = -2

II}:} i.§ .

-i ×

.

+ 2x.

-5×3 = 6

Ax = b

A = f ?I.E) HR" '

b = [f) ER" '

x= ER'

Addition element - wise

X = ( x.

,X

, ,. . .

xd ) y = ( y , , 42 ,. . - ya )

×, geld?! $ lmathbbcr)^dd$

Z = Xx y = ( x, -1g ,

,Xz -192

,. . .

,Xd -1yd ) END

^

ga..

- ¥?Z - * s

¥*jR ?

A,

B e IRnxd

C = At B ⇒ Ci;

- Ais + Be ;

A ⇒ . EI:De. a .IE#:::iT

MultiplicationHEIR

"dBe 112dam

C = AB EIRn 'm

Cii . E÷ainEf¥!!q:* ''

#re#coliA#wBA

, B. ( matrices AB legal maybe BA not legalnative CAB) c = A CBC ) ( unless a- m )

distributive ACB = ABT AC

not commutative AB # BA

G - fi, ⇒ tip

"

B = f ? I ? ] ER 'D

OBE:::::3( GB )

, ,

= I . 2 -13.4

2 t 12 = 14

-

B = a A Ai Be Rmdx EIR ← scalar

Be ; = a . Ai ; element wise

vector-veeformottipli.at#inner product and o - terprodoef

X EIR "

y EIR"

column vectors

outer productok if n # m

× . a " "

C- 112mm

lnnerldotlprod.deX

, g E Rd column vectors

x-y.x.ge 'T:!gJ" Iea. . .

. !4aigieR

output is scalar

associative,

distributive,

commutative

X. g. Z END a EIR

Lax, gtz > = ale , gtz7-akx.pt# ⇒

DotprodoctI w-

- ( E. I ) Vale , D

airtight.li?sieo:s!Y

*•"'

e " ,

if leastID -

-11*11=1

)o

I

GR?

u'

= Cleo )then245=11TEND)#I

315 Lv, U' D= 2. I t ↳= 2

Matrix - vector Multiplication-

A EIR" ''d

× ⇐ Rd

y -- Ax C- 112"

A

I aioiizd

' axesx.

E:÷÷÷t*

^ N( Vectors )•

× If ?k

how bis .

I vector v - Cv,

,ve

,. . .

,Vd ) C- Rd↳?µq

, ,qµ,q , ,,u,,u,,...µ#,µq

Hv - xh --E¥di-x = Euclidean distanceftp..ee: .

. iii.'

si:" ⇒

I i = I

Kubo - III. a ] Hit