100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified Latest Edition $18.49   Add to cart

Other

Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified Latest Edition

 46 views  0 purchase
  • Module
  • Linear Algebra & Optimization for Machine Learning
  • Institution
  • Linear Algebra & Optimization For Machine Learning

Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified Latest Edition Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 978303040343...

[Show more]

Preview 9 out of 117  pages

  • September 13, 2024
  • 117
  • 2024/2025
  • Other
  • Unknown
book image

Book Title:

Author(s):

  • Edition:
  • ISBN:
  • Edition:
  • Linear Algebra & Optimization for Machine Learning
  • Linear Algebra & Optimization for Machine Learning
avatar-seller
TestBanksStuvia
34567849 96 5
5 4  

5

!"#
$ %&
#
 '
(
(




 011

,




45678679
688677566675

756

6869575696 68!"9789 #
$ %568&7
896'86(8
759 $)
* 7756+9
9&
,68 866-8. *#
) 77564,86896(6
8!5
7569 )#
/ 660817566%
7" /$
# !6
-
8%8
559756 #
2 &713
75756 2
4 ,8 6885 !7" 24
5,8 6885 6,9 4)
7756645
77566,9 5



 0111

,

789 
999

 
9


 




!"#$%&'$()"$*+,("#-$.$%&/$'0$)12,1$%#+$+%,1$"3$4+&5(1$%0$-1")$(1%($627$.$4'$2-$
"#(1"5"&%4$("$.$8 '0$%&/$6227$(1+$/"($9#"/:,($"3$.$4;'$%&/$.$8 ;'$2-$&+5%(2*+<$
=>?@ABC>DEF>EE>GHIJ.$2.$
4'2$'$KE>LMFABN>EFD>OKF>PBHDQHBDFJQCGRFD>SGKIF>HI>TRF>QLU
@ABNQFHDQNKTFQCRPBTFQDV>FA>FEBIC>E>FEEWKRDBNIBLMFAUX>LTBOQFAPBTFQDERDBQC
FABERGBIBLMFAY>FCQIIQVEFARFFABDBEKIF>E\ZU=>>\?[LFABEBTQLNTREBYQLBTRLKEBR
E>G>IRDRDMKGBLFFQEAQVFARFFABDBEKIF>E%4]%YVA>TA>ELBMRF>PBU
^_"&-2/+#$%$-2(:%(2"&$2&$)12,1$'":$1%*+$(1#++$̀%(#2,+-$a0$b0$%&/$_0$"3$-2c+-$0Zd\0$
\d0Z0$%&/$0Zd0Z0$#+-9+,(2*+4'<$
6%7ef:99"-+$'":$1%/$("$,"`9:(+$(1+$̀%(#2.$9#"/:,($ab_<$!#"`$%&$+332,2+&,'$9+#g$
-9+,(2*+0$)":4/$2($,"`9:(%(2"&%44'$̀%h+$̀"#+$-+&-+$("$,"`9:(+$=ab?_$"#$)":4/$
2($̀%h+$̀"#+$-+&-+$("$,"`9:(+$a=b_?i$
6j7ek3$'":$1%/$("$,"`9:(+$(1+$̀%(#2.$9#"/:,($_ab0$)":4/$2($̀%h+$̀"#+$-+&-+$("$
,"`9:(+$=_a?b$"#$_=ab?i$
@ABGR>LHQ>LF>EFQlBBHFABE>mBQCFAB>LFBDGBN>RFBGRFD>SREEGRIIREHQEE>OIB
>LQDNBDFQDBNKTBOQFATQGHKFRF>QLRIRLNEHRTBDBWK>DBGBLFEU[LFABTREBQCab_Y
>FGRlBEEBLEBFQTQGHKFBb_$C>DEFU[LFABTREBQC_ab$>FGRlBEEBLEBFQTQGHKFB
_a$C>DEFU@A>EFJHBQCREEQT>RF>P>FJHDQHBDFJ>EKEBNCDBWKBLFIJ>LGRTA>LBIBRDL>LM>L
QDNBDFQDBNKTBTQGHKFRF>QLRIDBWK>DBGBLFEU
nf1")$(1%($23$%$`%(#2.$a$opqrosto$a$u5 av$ 0$(1+&$%44$(1+$/2%5"&%4$+4+`+&(-$"3$(1+$
`%(#2.$%#+$w<$
xQFBFARFa$8 av$uZUyQVBPBDYFA>EGRFD>SRIEQTQLFR>LEFV>TBFABN>RMQLRIBIBGBLFE
QCa$QL>FEN>RMQLRIU@ABDBCQDBYFABN>RMQLRIBIBGBLFEQCa$GKEFOBZU
zf1")$(1 %($23$)+$1%*+$%$̀%(#2.$-%(2-3'2&5$a$u av$0$(1+&$3"#$%&'$,"4:`&$*+,("#$.0$)+$
5
1%*+$.$a.$uZ<$
v
xQ FBFARFvFABFvDRLEHvQEBvQCFABvETRIRD.v$a.$DBGR>LEKLTARLMBvNU@ABDBCQDBYVBARPB


.$v a.$u=.$ a.?$u.$ a$ .$u4.$a.U@ABDBCQDBYVBARPB\.$ a.$uZU
 0

, 78 
4 4






4!"#$
%&'()*+*,
4)*-.'(&/0-1/.''23*+1/44

4405
68 78#49#:8""#4
8 9!7$
%&';<'*;(1/'=>,'((1&'?*1,;=?3+1;>+;)*1;/-*(1&'(3?/@/31',A>,/<3)1(/@
)/+3?-(/@4*-<,/0(/@95 2 
49
 4595
5

B','C45 ;(1&'51&)/+3?-/@4*-<95;(1&'51&,/0/@95D/1'1&*11&''=>,'((;/-/-
1&',;E&1</'(-/1)&*-E';@0'?3+1;>+F4.FGH*-<<;I;<'9.FG5J*)&)/?>/-'-1
/@1&'(3?,'?*;-(3-)&*-E'<;-)+3<;-E1&'1&)/?>/-'-1C0&','1&'()*+;-E@*)1/,(
)*-)'+'*)&/1&',/315
K8 !78#49784LM9L L4L
 #N#"#O9
 #N# O8MP
777"! 8:":8:"$
Q@1','=>,'((;-E1&'?*1,;=>,/<3)1*(1&'(3?/@/31',A>,/<3)1(C0')*-()*+''*)&
I')1/,;-1&'/31',A>,/<3)11/3-;1A-/,?CL0&;+'>L3++;-E/31*()*+*,?3+1;>+'@/,1&'
/31',A>,/<3)1)/?>/-'-15%&'?*1,;)'(4*-<9)/-1*;-1&'('-/,?*+;R'<I')1/,(C
0&','*(MP)/-1*;-(1&'('()*+*,?3+1;>+'(5S-/1&',0/,<(C)/-(;<',1&')*('C0&','
0'&*I'1&'>,/<3)1;-1&'@/++/0;-E@/,?3(;-E1&'51&)/+3?-4/@4*-<1&'51&
,/09 /@9T 2 
49
 4595
5
U-')*-'=>,'((1&;(?*1,;=>,/<3)1;-1&'@/++/0;-E@/,?T
2  
49
 45 95V45 95
 




5W XYZ[Z\ ]4595
^' ),'*1'*<;*E/-*+?*1,;=M;-0&;)&1&'51&<;*E/-*+'-1,F;(X55*-<1&'-),'*1'
4L*-<9L*(1&'-/,?*+;R'<I',(;/-(/@4*-<9C,'(>')1;I'+F5
_8
# 7#88!#:
 8"! 7:"!7$`74
78# 8"!7$
U-+F,/0;-1',)&*-E'/>',*1;/-(*,','23;,'<1/)/-I',1;11/1&';<'-1;1F?*1,;=5
S->*,1;)3+*,C;-1&'1&;1',*1;/-C0';-1',)&*-E'1&'1&,/0/@40;1&0&*1'I',
,/0)/-1*;-(1&'1&,/0/@1&';<'-1;1F?*1,;=5Q>',?31*1;/-?*1,;=0;++*+0*F(
)/-1*;-(3)&*,/05%&;(?*1,;=)*-.','>,'('-1'<*(1&'>,/<3)1/@*1?/(18
'+'?'-1*,F,/0;-1',)&*-E'/>',*1/,(.F1,'*1;-E'*)&;-1',)&*-E'/>',*1;/-*(*
?*1,;=?3+1;>+;)*1;/-5
a8 #77!#8""bc"#"4#:8
7#8!#5 4 :"$  !#d7"!e
 0

, 5 5 
8
55
5

 5 5
 5 ! " 5
#$$%&'()*+),
8-('./%'(0&-/1'23+./1'4-0%$5%&'+-6'7'(63%-%.)/-+%&'
0)$36/+),-('7<'(63 %'289&.+.+:'0-3+'.,.+%&'7'(63%-%.)/6-%(.4%&-%
0('-%'+ ;%&'/ 
8.+%&'./='(+'), 8>)*'='(;<7'(,)(6+'4-0%$5%&'
+-6'(')(2'(./1)/%&'()*+),-+7'(,)(6+)/%&'0)$36/+),8
?@ABC " 5!"5D2 5EE4FG<H

F5I 2 5G5J 2KLMNOPQRKSTUVKTUVKWUXKYMZK[\OP]\KMZKPZ^ZPN\K
 _!55D5F2G2<EH
55"FG<6F2G2<
`'%:'-/5./='(%.:$'6-%(.4),+<.a'I22I
829<&'/;*'+'%F26F ;-/2G26Gb
8c<8
d%0-/:''-+.$5+&)*/%&-%FG6FG8
??ABe5!!
5
 5  !f
ghi<5
5" !"
! "
 5 5
ZUNKMjj\YNKNT\K^ZMkKO\lXkNR
g_hi<5
5" !"! "
 5! "
YUkXLZKUm\OMNPUZK[U\lKZUNKMjj\YNKNT\K^ZMkKO\lXkNR
n5_5I_! 5 ! " 5 5
b-cod,5)3+%-(%*.%&%&'6-%(.4;%&'/%&'%*)+300'++.='()*)7'(-%.)/+0)(('+7)/2p
./1%)6-%(.0'+e8-/2eq0('-%'%&'6-%(.4eqe88r)%'%&-%6-%(.463$%.7$.0-%.)/
.+/)%0)663%-%.='-/2%&.+.+/)%%&'+-6'-+e8eq8s)('4-67$';()%-%.)/6-%(.0'+
2)/)%0)663%'*.%&+0-$./16-%(.0'+8t0-$./1%&',.(+%()*:5q,)$$)*'2:5./%'(p
0&-/1./1%&',.(+%-/2+'0)/2()*+0('-%'+-2.,,'('/%('+3$%%&-/%&')/'):%-./'2:5
('='(+./1%&'+')7'(-%.)/+8
b:cod/%&.+0-+';.,%&'()*-/20)$36/)7'(-%)(+-('e-/2e;%&',./-$('+3$%.+
ee8u'0-3+'),%&'-++)0.-%.=.%5),6-%(.463$%.7$.0-%.)/;bece-/2ebec
-('%&'+-6'89&'('+3$%,)$$)*+%&-%%&')(2'(2)'+/)%6-%%'(8
?vABE5
" 
 5 55!
"55"5
9&'('-('-,./.%'/36:'(),7'(63%-%.)/+),-+'w3'/0'89&'(',)(';-,%'(+)6'/36:'(
I),('7I'-%'27'(63%-%.)/+:5;%&'+'w3'/0'*.$$:'('7'-%'28d/)%&'(*)(2+*'
&-=' 6n8
?xABy5 5q!" 5!9567555

!55 !"z
 5!
5!!{5|bc5!5!55E5
" z
_!5!55|bc_"
5
 !9" 5!
9&'6-%(.47)$5/)6.-$0-/:'*(.%%'/-+7n}b56855
8c89&.+0-/:',3(%&'(
'47-/2'2-+,)$$)*+~
9 
9
7n }b5 c 5
8 67n } b8n }b55
qcc
5
68 56q
b9

67n } b8n }bqn } 55
qccc
5
60
+./1%&.+%57'),'47-/+.)/('03(+.='$5;)/'0-/):%-./%&'2'+.('2('+3$%8
 0

, ",79

9727
9 !9"7"#9$%&'$9(()!$'*+!) !9
 %)'$)'-$9.9(() $/0)
1(2-9(3%$'*
4566789:;<:85=;<><:?@A9A@8:8BCD:;99E:?@987D:;972 7><:?@A",7F@8:9G?7HIH@89
<?99J
J0J<EG%J:;9EH97K:<@E:;9D7FF7H@EL8M8:9>7D9N5<:@7E8O
97P
0Q
0
P %
7 Q
9
P
%"
09P %0 Q
R8@EL:;9D@?8::H79N5<:@7E8JH97K:<@E975%7QJH;@=;>9<E89@:;9?9 5%7?
9%BS7:9:;<:9% 4@8E7:6788@KF9K9=<589:;9:;@?G9N5<:@7EH75FGK9T@7F<:9GB
R8@EL9%JH9=<E9F@>@E<:9%:77K:<@E:;9D7FF7H@EL8M8:9>O
97P
0Q
79
"
90Q
4@E=99
@8E7EU9?7<EG90@8U9?7J@:>9<E8:;<:9=<EE7:K9U9?7J<EG0@8U9?7B
V7H9T9?J@D0@8U9?7J:;9E:;9D@?8:9N5<:@7E@>6F@98:;<:9@8U9?7@8H9FFBW;9?9D7?9J
H9?9<=;<=7E:?<G@=:@7EB
XYZ[Z\\]\^_[Z`a\Zbcad!199((()/9
(9 '9'!9!'3
).!'e39').!
'%').9199((()/9
'e39()!'3
).!'e39').'%9/)$9('*f!'
(9 9'9-0)%$g$
').-0)'9$%h*i) 3'-0)9(/
9)'!)
!g!'-0)%$g
3'!)(%*

W;9@G9E:@:M@8<8D7FF7H8O
7jjjj7P7jjhjj7jj5hjj7PjjPhjj7
kE9=<E9A6<EG:;9?@L;:I;<EG8@G9KM58@ELG7:6?7G5=:8J<EG:;9E<66FM:;9G@8I

:?@K5:@T96?769?:M:78;7H:;<:@:@89N5<F:7:;9F9D:I;<EG8@G9B
jj5hjj 7
PjjP hjj
7
6576hP h6hP 6P 76hP h6h
lD:9?=<E=9F@EL75::;9:9?>8@ET7FT@EL6h<EG=7E87F@G<:@EL7:;9?8JH9L9::;9
G98@?9G?985F:B
mnopqrsqtrsuovqswxyosqrozvsxwsqtrs{|}~xosr€|vpxsxwsqtrswx~~x‚pƒsyp„|op|qrswyqpxvs
9
)39†‡ˆ()/‰‡ˆ'$‰‡ˆ",‰‡-ˆ1*
@ŠF7L9P59,95597,797P59, 9
@@Š8@E9P59=7895 57978@E9559=789
@@@Š9"55979P599755990
@TŠ9A69P599A69P57979A69P599A69
‹Œ'!
3(-99d9g()19$')$)1)-%9e39%90911)
9)$).'$P
g$!-0$g).#g
Q#Q
sŽxuozsqt|qsqtpvs|€€oxpz|qpxs~xvrvspqvs| yo|}s
!$09'$/%'9$0.)
!)/$*
 0

,  
  





! 


 "" 



""
#$%& $
! '($$ $
) *+,""-) *+,""-.
 /& %.  01
!2)"3"45"6*+,""!*+,""'7 0.
 /%0#
-8
))*.$ 1
! 2)"*!*'
9:;<=>?@ABCDEFE@CGHCDCEB7IJEKDA
LA@A?AKAEMANCBO@CKKA?PEMMQEMRC@S
DE?B>JMK>
CAKHCDT>D!GAKHCUREMVD>OEOAMAK
E?BKHC?BAQABA?PEMMC?KDAC@O B3WA@FR@@
GHKHCF>MR[J?@>XLGAMMOCYD>RPHMZJRKREMM>DKH>P>?EMX>DMEDPCQEMRC@>XB>XKHC
>DBCD>X"3HA@KDAFIA@R@CBXDCURC?KMA?JEFHA?CMCED?A?PX>DDEVABMPC?CDEKA?P
KHCDE?B>JVD>\CFKA>?>XE??7BBEKEJEKDA
WE@W7)WL3
-$-#/$]$ /#& & "-$---0$
^SB./$$ #&B--&0]&$ _#-$]#$'`
%0#B8"[.$ $---0$]$ -"T4.-$ 
-$]#$]/$& 'a/&$$ -$& -$]#$
/$#-]$ 4 4$---0$.$& $ $$ /$ %
]$  //#&]]$ 3 "3""4-"3""4$%]]$'
& $ $$ 0/$>0> $%'b$/#.$ 
%]$ cd3c4-d"3e$%]]$'
9f;<=>?@ABCDKHCVCDKRDOCBJEKDA
g5)g! hi
GHCDCKHCQEMRC>XhA@@JEMME?Bg
i
EDCB7BJEKDAFC@3jH>GKHCX>MM>GA?PEVVD>
AJEKA>?k

g5T6gTThgTigT
HA@EVVD>
AJEKA>?A@R@CXRMGHC?gTA@EMDCEBI?>G?3

g5T
)
1g !hi2T)g1l 
! hgTi2
T )1l!hgTi2TgT
)1lThgTi!h,1gTi2,T33
32gT6gT ThgTigT
m/0$ $$ -#/$,g5$1gT3 hgTigT2-&$ -$$
& $]$&--$h. /#& -$]%%]' /
$/#/$ i0.#/ $/$& #&]
n/#&'
op;<jRVV>@CKHEK>RHEQCE07 0
JEKDA
g
A?GHAFHKHCD>G@^F>MRJ?@F>DDC@V>?BK>
VC>VMCA?E@>FAEM?CKG>DIA?KHC>DBCDq>H?
rED
qEFI
AJ
E?Bs>OA?3HCC?KD
1A
\2F>DDC@V>?B@K>KHC?RJOCD>XKAJC@VCD@>?A@C?KEJC@@EPCK>VCD@>?\3tuvwux
EJEKDA
L
@>KHEKLgLF>?KEA?@KHC@EJCA?X>DJEKA>?
ORKGAKHKHCD>G@^F>MRJ?@
A?KHC>DBCDrED
AJ
q>H?
s>OA?
E?BqEFI3
&#$$& $L
y""""z
8""""9
L){8""""9
""""|
""""

 0

, 89 
3 


!"
#$"



%& "



'(
)
"

*+#
,-./012213/041567-8 9:4:22821;4:52:3<./=>//8<.?:?8:42.848@84=./8A,-8=1448B
/91?<.?;28?;6-./ 55&55&C5&CCDE5F 7GHA
88 I"%**"*""**"#
J>991/86-:6K.//L55864.=:?<M./6-U8.?N84/ 810KA,-8?O38-:N8KMF MKFPA
,:Q.?;6-864:?/91/8O38 1 R 6 :.
? SK M T 
F SMK TU F PA
V/ .? ;/L 558 64L10
KO6-./.
/
6-8/:58://:L.?;6-:6MU KF KMU F P
AW?16-84314</OMUU ./6-8.?N84/810K:/
3822AX138N84O/.U?=86-8.?N84/8./>?.Y>8O38 5>/6-:N8MFMAZ2684?:6.N82LO38
=:?>/8KMUF KM F P
61://8466-:6KSMDMU TUFHA[8065>26.92L.?;RLMO38 ;86
SMKTSMDM TFHAJ.?=8MKF PO38-:N8MDMF HAW?16-84314</OMF MUA
8\ ]KGK5###K
'
^
*""
"

#U

%KGK5###K
"_*"#
[86M"R8;.N8?RLKGK5###K"A W6=:?R8/-13?.?<>=6.N82L6-:6M"./:2/1/64.=62L
64.:?;>2:4OR>63.6-:628:/6"`841413/:?<=12>5?/A,-8480148OM
3.22R86-8`841
5:64.@A
8a bcdeedfghijikglmfngnopk]KMq'"


Kr'*
"%*KMq#
I%

I"
"



'
)
"

*s
KM5 &Kq5F 5SKr5 & Mr5T
tI"


"*


''""
"


I"
"
"#
u>66-8N8468@10K:6:?<r:/6-814.;.?A,-8?6-8N846.=8/10M:?<q:48':?<


FD'A,-8?O6-8.<8?6.6L48<>=8/61/-13.?;6-8012213.?;v

wwD'ww5
  5
&ww&'wwF5Swwww5&
ww'ww5
T
   




,-././8:/L61/-13>/.?;<16941<>=6/AW?0:=6O6-8Z91221?.>/.<8?6.6L48<>=8/61
6-89:4:22821;4:52:3.?6-./=:/8x

8y z{|}~€‚~)"%.?684.14

'I
'("##



55GƒH

+
"*%77 ''
7'
#t 

I

"ST&STFG#q"
"

I"
%I"

'#U1991/.68

K
M"
#I%

I"

"



'
)
"

*s
  F 
"SKT "SMT
„  †5
,-8/.?8106-8:?;28./G D 7' A,-././8//8?6.:22L1R6:.?8<>/.?;/.?STF 
7GD=1/5STAJ.?=838:481?2L211Q.?;014.?684.14:?;28/O6-8/.?8./:91/.6.N8
Y>:?6.6LA,-8480148O38=:?.;?1486-891//.R.2.6L/.?STFD7  GD=1/5STA
 0

,   

26
6
 !  
  









6262 "   #
2 2

%&'$ 56 

$ (2
2q  
2_ _6%2r
'_

)*+**+  **
  ,#
  $    
%,' 56 





$ (2
q 2 
2_ _6%2r
'_

-  )*  
. 
727 6
77




$7
767




27 
*
 .
/01234567879:;4<=>5;?=@:A;74=BC6:D4BE=FGHIJKLM2NHJOPLQOGMR$S5TUVWVXLPLQOGM
P5JIG
O2JHLK
9MGO2OJHYRQGNHOLMZQ[GQ\]JIL2O2HY[L^5T2HKP_JIG
O2JHLK
9MGO2OJHY
RQ[GQ\]JIL2O2H2HY[L^_W`ILOXLMGO2OJGHa2OMJROGG
O2JHOXLQGGMKJH2OLIGbNHJO
PLQOGMIP52HKP_W`ILOXJIILONcOGIXG]OXLbG[[G]JHY]L[[Z\HG]HOMJYGHGaLOMJQ
JKLHOJO9d
QGI%^5e^_'$QGI%^5'QGI%^_'6IJH%^5'IJH%^_'

f.  )" 

S
%^5'T%^5'U 
S
%^_'T6%^_'U#

%^5'6%^5'5 
%6^_'6%6^_'5
%^5'
%^5'  T %6^_'
%6^_' 


.
%^5e^_' 
   

" 

V
++. *.
  

S5TU." 
+ *.
#

%^5e^_'$
%^5'
%^_'6%^5'%^_'

/g12h774i58B;:=6:79:;4<=>5;?=9B;45j=BC6:D4BE=FGHIJKLMOXLO]G[JHLI9$kRel
2HK9$mRe_JHOXLnZKJaLHIJGH2[c[2HLWoJHKOXLJHOLMILQOJGHGbOXLO]G[JHLI
9
]MJOJHYOXLLpN2OJGHIJHOXLbG[[G]JHYbGMabGM2ccMGcMJ2OL[9QXGILH&2HK
d

&R 9
 $


oJHKOXLJHOLMILQOJGHQGGMKJH2OLI%RT9'GbOXLO]G[JHLI
9JHPLMOJHYa2OMJR&W
 0

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller TestBanksStuvia. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $18.49. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

67096 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now

Start selling
$18.49
  • (0)
  Add to cart