fo da algebrajeffp/teaching/foda/lnotes/foda-l8.pdf · vector-veeformottipli.at# inner product and...
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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
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