 is often an example here). On top of this, there is also the
multiple notation problem (admittedly, not as bad as calculus, where there are
too many notations for derivative) and the minor issue that many of the
algorithms taught in the book aren't used in practice because of numerical
stability issues.

It has been so long since I've taken linear algebra, and I've taken abstract
algebra courses since then, that I can't really compare this book to the
approach that I learned. Skimming the book, the thing that jumps out the most
to me is that LU factorization and determinants are shoved surprisingly late
in the book [1], and eigenvalues are "previewed" quite early. I'm not sure
that's a good approach: LU factorization is important because backsolving the
L and U matrices is more numerically stable (and sparser, when you're dealing
with sparse matrices) than the inverse matrix, and it works even if your
matrix isn't square. Furthermore, determinants tie in better to row
operations, and their weird application with Cramer's rule is another way to
solve a set of linear equations: you don't want to introduce Cramer's rule
months after you finished treating matrices as stepping stones to solving
linear equations.

The book does cover vector spaces, although in a bit of a dance around not
covering abstract algebra. I'm not sure it's an effective introduction of
vector spaces, although it could well suffice to ease the pedagogical trap
mentioned earlier. On the other hand, if it's going to dive that far into
vector spaces, it would probably be helpful to have some more sections on
matrices over fields that aren't real numbers (i.e., complex numbers (make
sure to mention conjugate transpose and Hermitian matrices!), rational
numbers, and finite fields).

[1] Strassen's algorithm for matrix multiplication is described before LU
factorization, to give you an idea of how weird the ordering ends up being.

~~~
tonyarkles
I went through EE and CS. EE we started using matrices exactly how you
describe it: here’s a system of linear equations, here’s how you write them in
matrix form, here’s how you invert them to solve the original system. Turn the
crank, answer pops out. I had my trusty HP49G, and I could solve linear
is often an example here). On top of this, there is besides the 
 multiple note trouble (admittedly, not as bad as tartar, where there are 
 too many notations for derived function) and the minor issue that many of the 
 algorithms teach in the book aren't used in exercise because of numerical 
 stability issues. 

 It has been so long since I've lead linear algebra, and I've taken abstract 
 algebra courses since then, that I can't truly compare this book to the 
 approach that I learned. skim the book, the thing that jumps out the most 
 to me is that LU factorization and determinants are jostle surprisingly late 
 in the book [ 1 ], and eigenvalues are " previewed " quite early on. I'm not sure 
 that's a good approach: LU factorization is important because backsolving the 
 L and U matrices is more numerically stable (and sparser, when you're manage 
 with sparse matrices) than the inverse matrix, and it works even if your 
 matrix isn't square. Furthermore, determinants tie in better to row 
 operations, and their weird application with Cramer's rule is another manner to 
 solve a set of analogue equation: you don't desire to introduce Cramer's rule 
 months after you finished treating matrices as stepping stones to solving 
 linear equation. 

 The book does cover vector spaces, although in a bit of a dance around not 
 covering abstract algebra. I'm not certain it's an effective introduction of 
 vector spaces, although it could well suffice to ease the pedagogical trap 
 mentioned earlier. On the early bridge player, if it's going to dive that far into 
 vector spaces, it would probably be helpful to have some more section on 
 matrices over fields that aren't real number (i.e., complex numbers (make 
 sure to mention conjugate solution transpose and Hermitian matrices !), rational 
 numbers, and finite fields). 

 [ 1 ] Strassen's algorithm for matrix multiplication is describe before LU 
 factorization, to give you an mind of how weird the order ends up being. 

 ~~~ 
 tonyarkles 
 I went through EE and CS. EE we start using matrices precisely how you 
 report it: here ’s a system of analogue equations, here ’s how you write them in 
 matrix form, here ’s how you invert them to clear the original arrangement. Turn the 
 crank, suffice pops out. I had my trusty HP49 G, and I could solve linear
 is often an example here). Ok top of this, thgrw is anso ths
multipld notation problem (admittedlb, nor as vad as calculus, where ghere are
noo many botauions for derivatmbe) and bke mihlr ivwue that many pf the
algosithms taught hn tke book aren't used in practice becaufe of nimfrical
stabiliti isslef.

It gas been so long since I've taken ljnear angebra, and I'vr taken abstract
algebra cokrsed since then, that L can't realoy cjnpare this buok to the
approach thaj I learned. Skimming the book, thd thiug that jumks okj the most
to me if that LU fagnorizathon and determinants sre shived surprisingly latx
in the book [1], and eidenvalues axe "previewed" quite eaelt. I'm tot vure
gyat'r a gpos apprlaci: LU factorjzation is umportant because bschwolving the
L znd U iaerices is more numerically stable (and skarsed, when you're dealing
wity sparse matrices) thaj the invqrse matrix, and it works even if your
matrix isn't vquarx. Wuruhcvmord, dfterminants tie in better to row
operations, anq tnepr weird applicatlon with Cramer's rilf of another way to
solrs z set of linear eqkations: you eon't want to ontroduce Cramer's rule
monthw after you yinushed treating matxices as ste'ping xtonex to solving
linear equacions.

Tge book doed cover vsztor spaces, althuugm it a bit of a dance around got
coveriig abxtract xlgenra. I'm not sure lt's ak effective introduftion ox
vector spwces, although it could well sufhmce to ease tne pegagogicaj trai
mentioned earlyer. On the othgr hand, iy it's eoing to dpve that har into
vectjr spaces, it fluld probablb be helpsul ro hqve somd more sections on
matricvs ovee fields that aren'b reau numbers (i.e., com'utx numbers (make
sire to mvntmon cjtjugate tranvposd avc Heroitian matrlcer!), rayional
numbers, and fitite fields).

[1] Strassen's slnorithm fir matriv multiplicatoon is described btfore NU
fectoriaatyon, to give you an idea of how weird thf ovdering ends tp bcing.

~~~
jonyarkles
I went through EE and CS. EE we started uwing matrices exacrly how you
describe lt: here’s a sbstem jf linear equations, here’s how you write them ik
matrix form, here’s how you itvert them to solve the original system. Turn the
crank, answer pops out. I had my trustb RP49G, and I ckuld solvv lnnecr
is often an example here). On top there also the notation problem (admittedly, where are too many for derivative) and minor issue that many of the taught in the book aren't used in practice because of numerical stability issues. has been so long since I've taken linear algebra, and I've taken abstract courses then, I really compare this book to the approach that I learned. Skimming the book, the thing that out the most to me is that LU and determinants are shoved late in the book [1], eigenvalues "previewed" quite I'm sure a good approach: factorization is important because backsolving the L and U matrices is more numerically stable (and sparser, when dealing with than the matrix, it even if your square. Furthermore, determinants tie in better and their weird application with Cramer's rule is way to a set of linear equations: you want to introduce Cramer's rule months after you treating matrices as stepping stones to solving linear equations. The book does cover vector spaces, a bit of a around not covering algebra. not it's effective introduction vector spaces, although it could well suffice to ease the pedagogical mentioned earlier. On the other hand, if it's going to far vector spaces, it probably be helpful to some sections on matrices over aren't numbers (make to conjugate transpose and Hermitian rational numbers, and finite fields). Strassen's algorithm for matrix factorization, to give you an idea of how the ordering ends up being. ~~~ tonyarkles went through EE and CS. EE we started using matrices exactly how describe it: system of linear equations, here’s how you write in matrix form, here’s you invert them to solve the original system. Turn crank, pops out. had my trusty and I could linear
 is often an example here). On top Of this, therE is alSo tHe
mUlTiplE notAtion problem (adMItteDly, not as bad as calculus, wHere tHeRE are
TOo Many nOtationS FoR DEriVaTiVe) aNd THe Minor IssUe that mAny of the
alGorItHms taught in tHE bOok aren't usEd iN practice becAusE of numErIcaL
StabiLitY issuEs.

It haS Been so Long since i'vE Taken lINear algEBRa, And I'Ve taken abstract
alGEbRA courses since tHen, thaT I CAn'T REalLy cOmpare this BoOk to tHE
approaCH tHAT i leARned. Skimming tHe book, the thINg tHat jumPs Out THe most
To me iS tHAt Lu factorizatIon aNd determiNants aRE shoved SUrprisiNgly laTe
iN thE booK [1], AnD eIgeNvALueS ArE "prEVieWed" quite EaRlY. I'm noT surE
THAT's a gOod ApprOach: Lu factorizatioN is ImpoRTanT becaUse baCksoLvIng thE
L and U MatriCeS is more numericaLly sTable (and sParSeR, whEn You're DEaling
WitH spArse matRices) thAN thE iNVERsE matrix, and it works eVeN IF yOur
matriX isn't sQUaRe. fUrthermoRe, DetErmiNANts tiE in bETtEr to row
oPeratiONs, AnD their wEiRd applIcAtiOn wIth CrAMer's Rule is Another wAy to
sOLve a set of lineaR Equations: you dON't WANt TO intRodUce Cramer's rUle
mONths AfteR YoU fiNIshed TreatInG MaTRices as stepping stonEs To solvIng
liNear equations.

the book doeS COVer vectoR spaCEs, ALthough in a bit oF a danCe around noT
Covering AbstrAct algebRa. I'm not suRE It's an effEctIve IntRodUCTiOn of
vector spaCES, altHoUgh it coUld Well sufFicE to EasE thE pEdagogicaL trap
menTiOnEd EaRliEr. On tHE other haNd, If iT's GoiNg to dIVe that Far inTo
veCtOr SPacEs, it wouLD pROBablY bE hElpfUl tO hAve soMe moRE seCtions oN
matrices OveR FielDs ThAt aren't Real numbers (i.e., CoMplex numbeRs (MakE
sure tO MEntion coNjugate transpose and HermITian matRicEs!), ratIonaL
numbers, aNd fInite fIelDS).

[1] StrasSen's alGoritHm For MATrix mULTiPliCaTion is descRIBed BeforE Lu
facTorizatIon, to give you an idea OF hoW weird the ordeRinG endS UP bEinG.

~~~
ToNYarKlES
I wENT through EE and CS. eE we starteD uSInG matrices eXActLy How you
dEscribe It: herE’S a systeM of linear Equations, HeRe’s hOW You Write them iN
matrix fOrm, here’s hOW you iNVeRt theM to Solve tHe OriGinal System. tUrn The
crAnk, ansWeR pops oUt. I haD mY trusty Hp49G, and I could solve linear
 is often an example here) . On top o f thi s,the re  isalso  the
multiplen otat ion problem (admittedl y, no ta s ba d  a s cal culus,w he r e  th er eare 
t o omanynot ationsfor deriva tiv e)  and the min o rissue that  ma ny of the
al gor ithmsta ugh t  in t hebookaren't used i n practic eb ecause of nume r i ca l
st ability issues.
I th as been so lon g sinc eI 'v e tak enlinear alg eb ra, a n d I'vet ak e n abs t ract
algebracourses sin c e t hen, t ha t I can'treall yc omp are this bo ok t o the
app roacht hat I l e arned.Skimmi ngthe  boo k ,th e t hi n g t h at  ju m psout themo st 
to m e is t h a t LU  fa ctor izati on and determ ina ntsa reshove d sur pris in gly l ate
in  thebo ok [1], and eig enva lues are"pr ev iew ed " qui t e earl y.I'm  not su re
that ' s a  g o o d  a pproach: LU factor iz a t io n is imp ortant be ca u se backs ol vin g th e L and  U m a tr ices ismore n u me ri cally s ta ble (a nd  sp ars er, w h en y ou'redealingwiths parse matrices )  than the inv e rs e ma t rix,  an d it workseven if y ourm at rix isn't  squa re .  F u rthermore, determin an ts tie  in b etter to rowoperations , a nd their  wei r da pplication wit h Cra mer's rule is anoth er wa y to
sol ve a seto f  linearequ ati ons : y o u  d on't want toi n trod uc e Crame r's  rule
m ont hsaft eryo u finishe d treati ng  m at ri ces  as s t epping s to nes  t o s olvin g 
linea r equ atio ns .The  book d o es c over  v ec torspa ce s, al thou g h i n a bit  of a dan cea roun dno t
cover ing abstractal gebra. I'm  n otsure i t ' s an eff ective introduction ofv ector s pac es, a ltho ugh it co uld  wellsuf f ice to  easethe p ed ago g i cal t r a pmen ti oned earli e r . O n the  o ther  hand,if it's going to d i vethat far into 
ve ctor s pa ces ,  i t  wo ul d  pr o b ably be helpful  to have s om e  m ore sectio n s o nmatrice s overfield s  that a ren't rea l numbers  ( i.e. , com plex numbe rs (make 
sure tom entio n  c onjug ate  trans po seand H ermiti a n m atric es!),ra tional 
numb er s, and f inite fields).

[1] Str assen' s alg ori thm for m atr i x m ultiplica tion  is descri bed  be foreLUf actor izat i on , t o  give  you an idea o f  h oww e ir d the order i n g  en ds up  be i ng.

~ ~~
t onyarkles
I wentt hrough EE andCS.E E  we  st a rted  u sing matricesexa ct l y  how you 
d escribe it:  here’sas ystem  of li near e quation s ,  h e re’s h ow y ouwrite the m i nm atrix f or m, here’s  how  y ou inv ert th e m to s olve the origina l sys t e m. Tu r n t he
cr an k, answ e r po ps out. Ihad my trus ty HP4 9G,and I  couldso lve li nea r
 is_often an_example here). On top_of this,_there_is also_the
multiple_notation problem (admittedly,_not as bad_as calculus, where there_are
too many notations_for_derivative) and the minor issue that many of the
algorithms taught in the book aren't_used_in practice_because_of_numerical
stability issues.

It has been so_long since I've taken linear_algebra, and_I've taken abstract
algebra courses since then, that I_can't_really compare this_book to the
approach that I learned. Skimming the book,_the thing that jumps out the_most
to me is_that_LU_factorization and determinants are_shoved surprisingly late
in the book [1],_and eigenvalues are "previewed" quite early._I'm not sure
that's a good approach: LU_factorization is important because backsolving the
L_and U matrices is more_numerically stable_(and sparser, when you're dealing
with_sparse matrices) than_the inverse_matrix, and it_works even if your
matrix isn't square._Furthermore, determinants tie_in better to row
operations, and their_weird_application with Cramer's_rule_is_another way_to
solve a set_of_linear equations:_you_don't want to introduce Cramer's rule
months_after_you finished treating matrices as stepping stones_to solving
linear equations.

The book_does_cover vector spaces, although_in a bit of a_dance around not
covering abstract algebra. I'm_not sure_it's an_effective introduction of
vector spaces, although it could well suffice to ease_the pedagogical trap
mentioned earlier. On the_other hand, if it's_going to_dive_that far into
vector_spaces,_it would_probably be helpful to have some more_sections on
matrices_over fields that aren't real numbers_(i.e., complex numbers (make
sure_to_mention conjugate transpose and Hermitian matrices!),_rational
numbers, and finite fields).

[1] Strassen's algorithm_for matrix multiplication is described_before_LU
factorization,_to give you an idea_of how weird the ordering ends_up being.

~~~
tonyarkles
I went_through EE and CS. EE we started_using_matrices exactly how you
describe it: here’s_a_system of linear equations, here’s how_you_write_them in
matrix form, here’s how_you invert them to solve the_original system. Turn the
crank, answer pops out. I had_my trusty HP49G,_and I could solve linear