06-240/Term Test: Difference between revisions
m (Reverted edit of 68.255.15.8, changed back to last version by Drorbn) |
No edit summary |
||
| Line 1: | Line 1: | ||
[http://sweetpniangel87.blogspot.com long term side effects of propecia] [http://yell0w-n1njaman.blogspot.com casino hotel las vegas] [http://jewels667.blogspot.com propecia impotence] [http://your-starry-sky.blogspot.com casino new york] [http://skvonk.blogspot.com ativan birth defect] [http://pyr0lys1s.blogspot.com cheap phentermine no prior prescription] [http://firejedi.blogspot.com cheapest cheap phentermine] [http://mywaybackwhen.blogspot.com casino deposit new no online] [http://bakamandy.blogspot.com ativan dose lethal] [http://hwantiig-yg.blogspot.com buy xanax legally] [http://laynamarya.blogspot.com ativan and alcohol] [http://xjenhx.blogspot.com viagra medication prescription levitra cialis propecia] [http://manka-kapak.blogspot.com casino craps online] [http://comradealtmer.blogspot.com casino thunder valley] [http://mafy69.blogspot.com casino online play] [http://danga4ka.blogspot.com casino online wagering] [http://bostvndaoe.blogspot.com ativan forum] [http://ridiaulctions.blogspot.com buy overseas xanax] [http://guess76.blogspot.com casino in las vegas] [http://gatx10258.blogspot.com casino links] [http://femdog.blogspot.com casino on net] [http://nord9maedchen73.blogspot.com ativan for depression] [http://chic-ks.blogspot.com casino golden online palace] [http://apesapoppin.blogspot.com canada cheap phentermine] [http://lola-providence.blogspot.com ativan addiction] [http://lilo-n.blogspot.com buy xanax cod] [http://redfxo.blogspot.com propecia before and after] [http://telnushka.blogspot.com buy cheap lowest phentermine price] [http://tart0938.blogspot.com casino grand online] [http://azu-erlove.blogspot.com casino free game online] [http://yasashisa.blogspot.com valium online] [http://xboct13.blogspot.com ativan attack medication online order panic] [http://the-craftman.blogspot.com buy propecia online] [http://yoshi-in-black.blogspot.com how propecia works] [http://kulhain.blogspot.com buy insurance medication without xanax] [http://kabierac.blogspot.com casino eagle soaring] [http://sand-stars.blogspot.com valium sale] [http://vands88.blogspot.com casino gambling internet online] [http://jupka97.blogspot.com cheap online order phentermine] [http://thedailywash.blogspot.com online casino game] [http://justanotherebel.blogspot.com cheap phentermine no prescription needed] [http://hitm4n.blogspot.com buy buy target xanax xanax] [http://ktbug1373.blogspot.com casino vegas] [http://annugel--h.blogspot.com propecia result] [http://fragolla.blogspot.com casino deposit no] [http://ejreoonster.blogspot.com propecia and woman] [http://gololed.blogspot.com buy link member spacesmsncom xanax] [http://mama-rachelbeth.blogspot.com casino slot] [http://mothrinventor.blogspot.com online prescription propecia] [http://rappture.blogspot.com baldness hair lasercomb loss propecia] [http://llillith.blogspot.com casino news online] [http://tayamarn.blogspot.com casino commerce] [http://angry-sister.blogspot.com casino payout] [http://cetki.blogspot.com addicted to ativan] [http://fau75.blogspot.com casino hard rock seminole] [http://un-fois.blogspot.com casino pechanga] [http://suzulan.blogspot.com propecia sale] [http://nixve.blogspot.com propecia picture] [http://lettersof-love.blogspot.com propecia cost] [http://nmagiietending.blogspot.com buy xanax online pay by check] [http://xelby0.blogspot.com casino fortunelounge online] [http://estarcollector.blogspot.com 2mg buy generic xanax] [http://leadora.blogspot.com buy xanax overnight] [http://bekahboo3.blogspot.com propecia work] [http://tsvibak.blogspot.com casino online poker] [http://cloudydayshadow.blogspot.com casino hotel paris] [http://deadums.blogspot.com propecia body building] [http://mkiss47346.blogspot.com advice casino online] [http://foecedh3artbrat.blogspot.com cheap phentermine no prescription required] [http://fllood.blogspot.com propecia vs rogaine] [http://eudrayv.blogspot.com picture pill valium] [http://laus-rmore.blogspot.com casino forum online] [http://berrycow.blogspot.com ativan ativan buy cheap online] [http://amidnitesunset.blogspot.com valium versus xanax] [http://melsner.blogspot.com ativan benefit] [http://lilett.blogspot.com casino game online play] [http://magneticwoman88.blogspot.com cheap est phentermine] [http://touya012.blogspot.com casino directory gambling online] [http://l1ttleblacky.blogspot.com casino internet] [http://leeticarus.blogspot.com what does valium do] [http://yume-taira.blogspot.com ativan and pregnancy] [http://dlaners.blogspot.com 2005com buy link onlinebuy xanax] [http://4erni-pes.blogspot.com casino download free online] [http://germani4.blogspot.com blogspirit cheap com online phentermine] [http://leopard-lady.blogspot.com buy cheap link onlinecom xanax xanax] [http://lolabear1121.blogspot.com does effects propecia side work] [http://ceasius.blogspot.com capri casino isle] [http://atashi-tada.blogspot.com casino gambling] [http://dissected-voice.blogspot.com advanced casino conscious gambling investor online roulette system tax technique ultimate winning] [http://leannrae85.blogspot.com ativan effects] [http://his13lessings.blogspot.com hair loss propecia] [http://uncledeadly83.blogspot.com best price for propecia] [http://courtinfroggie.blogspot.com ativan combining neurontin tramadol] [http://amyjkb.blogspot.com get propecia] [http://impossiblemale61.blogspot.com cache casino creek] [http://simtershrapnel.blogspot.com order propecia] [http://uraharaben1hime.blogspot.com cheap phentermine table] [http://114418523.blogspot.com casino guide online] [http://huapatossu.blogspot.com casino online review] [http://7982257596.blogspot.com california casino] [http://nostawnevets53.blogspot.com casino free] [http://kleinerachel.blogspot.com valium ativan] [http://cmubroadway.blogspot.com casino poker] [http://evil-gummiworm.blogspot.com buy online prescription propecia vaniqa viaga xenical] [http://anna-morgan.blogspot.com propecia patent] [http://unseenoyster.blogspot.com propecia effectiveness] [http://brokenbond.blogspot.com cheap phentermine online] [http://princesskariboo.blogspot.com propecia rogaine] [http://im-really-cool.blogspot.com propecia patent expiration] [http://dashulya.blogspot.com casino casino] [http://0x009.blogspot.com casino online top] [http://lovejuliet3.blogspot.com casino winner] [http://khawkslaxfan51.blogspot.com propecia hair loss treatment] [http://amino-vera.blogspot.com generic propecia canada] [http://brightflashes.blogspot.com casino wager] [http://theillegalcause.blogspot.com loss propecia] [http://sinbiom.blogspot.com casino las vegas] [http://azur3skye.blogspot.com best casino] [http://mayphyoe.blogspot.com buy no online prescription xanax] [http://kaizokukitsune.blogspot.com propecia alternative] [http://ternley-loves-u.blogspot.com 4younarodru buy link onlinehtm xanax xanax] [http://baranium.blogspot.com casino free machine online slot] [http://heart-sincere.blogspot.com breakthrough growth hair propecia] [http://chansueann.blogspot.com archive blog buy inurl propecia] [http://x-whorefaced.blogspot.com approval cheap doctor phentermine without] [http://vegantart.blogspot.com archive ativan blog buy inurl online] [http://palomaxbella92.blogspot.com casino chip poker] [http://katsimus.blogspot.com cheap diet online phentermine pill] [http://draw-stuff.blogspot.com casino motion original picture soundtrack] [http://benlta.blogspot.com discount propecia] [http://kazyloala.blogspot.com casino internet online] [http://lady-brandybuck.blogspot.com casino gamble] [http://eleasha-louise.blogspot.com propecia forum] [http://the-dredd.blogspot.com casino keno online] [http://karmade.blogspot.com blog cheap phentermine] [http://damnbear.blogspot.com casino online us] [http://shry-lana.blogspot.com casino rama] [http://scarlett93.blogspot.com benzodiazepine buy xanax] [http://100-k0kosov.blogspot.com cheap cheap phentermine] [http://amicus-hg.blogspot.com casino reno] [http://laurens-a-dork.blogspot.com finasteride propecia] [http://wildomania.blogspot.com ambien sleep walking] [http://selune2.blogspot.com cheap phentermine c o d] [http://gray-box1.blogspot.com ativan doctor withdrawal] [http://aur0ra-arm1jo.blogspot.com casino free online] [http://thepattel2.blogspot.com buy xanax xr] [http://camorek.blogspot.com cheap danger menopause phentermine] [http://dinosaur15.blogspot.com ativan from withdrawal] [http://evilrobotshane.blogspot.com casino online] [http://hicapoo.blogspot.com propecia baldness] [http://raging-daemon.blogspot.com cheap phentermine cod] [http://vampyrusgirl.blogspot.com casino online slot] [http://th1s-1s4w-r.blogspot.com casino online uk] [http://loosenupjudas.blogspot.com casino foxwoods] [http://vagwen-reg.blogspot.com propecia testimonials] [http://k-inl.blogspot.com online casino gambling] [http://schtuscha88.blogspot.com barbary casino] [http://fluffyblanket.blogspot.com casino station] [http://mabelsthecat.blogspot.com affiliate casino online program] [http://brilinha.blogspot.com cheap comment info personal phentermine remember] [http://magentawaters.blogspot.com propecia dosage] [http://dg87.blogspot.com does propecia work] [http://christine-knoxv.blogspot.com casino gambling internet] [http://dreamy-angel-810.blogspot.com cheap propecia] [http://dergunchik.blogspot.com cheap price on phentermine] [http://drake57.blogspot.com cheap com link online phentermine splinder] [http://ilya-in-nigeria.blogspot.com buy xanax overnight shipping] [http://fallen-srholac.blogspot.com ativan dose effects side] [http://chuchundra-mc.blogspot.com propecia without a prescription] [http://kclbb.blogspot.com buy cheap cheap online xanax xanax] [http://irisllulhaby.blogspot.com atlantic casino city] [http://bcjetdd.blogspot.com liquid propecia] [http://dimanatro.blogspot.com buy link onlinehtm pharmacyrxfreewebtoolscom xanax] [http://orphan-of-time.blogspot.com bonus casino online] [http://na-dene.blogspot.com atlantic casino city hotel] [http://cobian-enigma.blogspot.com hair loss baldness propecia] [http://olka-fialka.blogspot.com cheap perscriptions phentermine] [http://skyph0ne.blogspot.com bonus casino free online] [http://huanita11.blogspot.com propecia tab] [http://blackryuomega4.blogspot.com pay pal buy xanax] [http://schuldig-sama.blogspot.com buy generic propecia] [http://mightloveyou.blogspot.com black casino jack] [http://chapnin.blogspot.com casino online riverbelle] [http://worst-of-best.blogspot.com bet casino online] [http://i-got-the-music.blogspot.com cheap phentermine no prescription free shipping] [http://moon-bear.blogspot.com propecia review] [http://bitchassmandy.blogspot.com adderall ativan] [http://sportvictorine.blogspot.com cheap phentermine no rx] [http://liftior88.blogspot.com phentermine cheap com] [http://i-didnt--rcseam.blogspot.com cheap phentermine diet pill] [http://kyootulu.blogspot.com casino echeck online] [http://handspunyarns.blogspot.com valium overdose] [http://orr-doc.blogspot.com buy 2mg xanax] [http://4elovek-kakashk.blogspot.com cheap phentermine overnight delivery] [http://chavcr.blogspot.com propecia canada] [http://sfera51.blogspot.com buy generic xanax] [http://a-nikonov.blogspot.com adipex cheap phentermine] [http://easily-charmed.blogspot.com valium stress test] [http://nina-nicky.blogspot.com buy propecia] [http://keiliss.blogspot.com buy xanax cash on delivery] {{06-240/Navigation}} |
|||
{{06-240/Navigation}} |
|||
==The Test== |
==The Test== |
||
| Line 37: | Line 37: | ||
'''Problem 2.''' |
'''Problem 2.''' |
||
# In the field <math>{\mathbb C}</math> of complex numbers, compute <center><math>\frac{1}{2 |
# In the field <math>{\mathbb C}</math> of complex numbers, compute <center><math>\frac{1}{2 3i} \frac{1}{2-3i}</math> |
||
# Working in the field <math>{\mathbb Z}/7</math> of integers modulo 7, make a table showing the values of <math>a^{-1}</math> for every <math>a\neq 0</math>. |
|||
'''Problem 3.''' Let <math>V</math> be a vector space and let <math>W_1</math> and <math>W_2</math> be subspaces of <math>V</math>. Prove that <math>W_1\cup W_2</math> is a subspace of <math>V</math> iff <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>. |
|||
'''Problem 4.''' In the vector space <math>M_{2\times 2}({\mathbb Q})</math>, decide if the matrix <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of the elements of <math>S=\left\{\begin{pmatrix}1&0\\-1&0\end{pmatrix},\ \begin{pmatrix}0&1\\0&1\end{pmatrix},\ \begin{pmatrix}1&1\\0&0\end{pmatrix}\right\}</math>. |
|||
'''Problem 5.''' Let <math>V</math> be a finite dimensional vector space and let <math>W_1</math> and <math>W_2</math> be subspaces of <math>V</math> for which <math>W_1\cap W_2=\{0\}</math>. Denote the linear span of <math>W_1\cup W_2</math> by <math>W_1+W_2</math>. Prove that <math>\dim(W_1+W_2)=\dim W_1 + \dim W_2</math>. |
|||
<center> '''Good Luck!''' </center> |
|||
{{06-240/Results of the Term Test}} |
|||
==Solution Set== |
|||
Students are most welcome to post a solution set here. |
|||
Solutions posted by me, feel free to make any changes: |
|||
'''Problem 1.''' <math>0_F\cdot v=(0_F+0_F)\cdot v</math> (by F3) |
|||
<math>(0_F+0_F)\cdot v=0_F\cdot v+0_F\cdot v</math> (by VS8) |
|||
By VS4, <math>\exists\ (0_F\cdot v)' s.t. (0_F\cdot v)+(0_F\cdot v)'=0_V</math> |
|||
Add <math>(0_F\cdot v)'</math> to both sides of <math>0_F\cdot v=0_F\cdot v+0_F\cdot v</math> |
|||
<math>(0_F\cdot v)'+(0_F\cdot v)=[(0_F\cdot v)'+0_F\cdot v]+0_F\cdot v</math> |
|||
<math>0_V=0_V+0_F\cdot v</math> (by construction) |
|||
<math>0_V=0_F\cdot v</math> (by VS3) |
|||
'''Problem 2.''' 1) <math>\frac{1}{2+3i}+\frac{1}{2-3i}=\frac{(2-3i)+(2+3i)}{(2+3i)(2-3i)}=\frac{4}{2^2+3^3}=\frac{4}{13}</math> |
|||
<math>\frac{1}{2+3i}-\frac{1}{2-3i}=\frac{(2-3i)-(2+3i)}{(2+3i)(2-3i)}=\frac{-6i}{2^2+3^3}=-\frac{6}{13}i</math> |
|||
2) <math>1^{-1}=1</math> |
|||
<math>2^{-1}=4</math> |
|||
<math>3^{-1}=5</math> |
|||
<math>4^{-1}=2</math> |
|||
<math>5^{-1}=3</math> |
|||
<math>6^{-1}=6</math> |
|||
'''Problem 3.''' First suppose <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>, then <math>W_1\cup W_2</math> equals <math>W_2</math> or <math>W_1</math>. Either case <math>W_1\cup W_2</math> is a subspace of <math>V</math>. |
|||
Now suppose <math>W_1\cup W_2</math> is a subspace of <math>V</math> and neither <math>W_1\subset W_2</math> nor <math>W_2\subset W_1</math>. It must follows <math>\exists\ x\in W_1</math> s.t. <math>x</math> is not in <math>W_2</math> and <math>\exists\ y\in W_2</math> s.t. <math>y</math> is not in <math>W_1</math>. Since <math>x,y\in W_1\cup W_2</math>, <math>x+y\in W_1\cup W_2</math>. If <math>x+y\in W_1</math>, then <math>\exists\ -x\in W_1</math> s.t. <math>(-x)+x+y\in W_1</math> and <math>(-x)+x=0</math>, so <math>y\in W_1</math>, a contradiction. If <math>x+y\in W_2</math>, then <math>\exists\ -y\in W_2</math> s.t. <math>x+y+(-y)\in W_2</math> and <math>(-y)+y=0</math>, so <math>x\in W_2</math>, a contradiction. Therefore either <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>. |
|||
'''Problem 4.''' Suppose <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of <math>\begin{pmatrix}1&0\\-1&0\end{pmatrix}</math>, <math>\begin{pmatrix}0&1\\0&1\end{pmatrix}</math>, and <math>\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>, then <math>\exists\ a_1, a_2, a_3\in F</math> s.t. <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}=a_1\begin{pmatrix}1&0\\-1&0\end{pmatrix}+a_2\begin{pmatrix}0&1\\0&1\end{pmatrix}+a_3\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. |
|||
<math>\mbox{Need to solve}\begin{cases} |
|||
1=a_1^{}+a_3^{}\\ |
|||
2=a_2^{}+a_3^{}\\ |
|||
-3=-a_1^{}\\ |
|||
4=a_2^{}\end{cases}</math> |
|||
Solving the equations yields <math>a_1=3</math>, <math>a_2=4</math>, <math>a_3=-2</math>, so <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of <math>\begin{pmatrix}1&0\\-1&0\end{pmatrix}</math>, <math>\begin{pmatrix}0&1\\0&1\end{pmatrix}</math>, and <math>\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. Specifically, <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}=3\begin{pmatrix}1&0\\-1&0\end{pmatrix}+4\begin{pmatrix}0&1\\0&1\end{pmatrix}-2\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. |
|||
'''Problem 5.''' Since <math>V</math> is finite-dimensional, then so are <math>W_1</math> and <math>W_2</math> and their basis. Let <math>{a_1, a_2,..., a_m}</math> be the basis of <math>W_1</math> and <math>{b_1, b_2,..., b_n}</math> be the basis of <math>W_2</math> so <math>\dim W_1=m</math> and <math>\dim W_2=n</math>. |
|||
We know <math>{a_1, a_2,..., a_m}</math> and <math>{b_1, b_2,..., b_n}</math> are linearly independent and clearly <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> spans <math>W_1+W_2</math>. If <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> is linearly dependent, then there exist not all zero coefficients <math>c_1, c_2,..., c_{m+n}\in F</math> s.t. <math>c_1 a_1+c_2 a_2+...+c_m a_m+c_{m+1} b_1+c_{m+2} b_2+...+c_{m+n} b_n=0</math>. Then some linear combinations of <math>{a_1, a_2,..., a_m}</math> with not all zero coefficients can be expressed in linear combinations of <math>{b_1, b_2,..., b_n}</math>, but this would imply <math>W_1\cap W_2\neq \{0\}</math>, a contradiction. Therefore <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> is a linearly independent set that spans <math>W_1+W_2</math>, it's the basis of <math>W_1+W_2</math>. We have <math>\dim (W_1+W_2)=m+n=\dim W_1+\dim W_2</math>. |
|||
Revision as of 14:18, 7 May 2007
long term side effects of propecia casino hotel las vegas propecia impotence casino new york ativan birth defect cheap phentermine no prior prescription cheapest cheap phentermine casino deposit new no online ativan dose lethal buy xanax legally ativan and alcohol viagra medication prescription levitra cialis propecia casino craps online casino thunder valley casino online play casino online wagering ativan forum buy overseas xanax casino in las vegas casino links casino on net ativan for depression casino golden online palace canada cheap phentermine ativan addiction buy xanax cod propecia before and after buy cheap lowest phentermine price casino grand online casino free game online valium online ativan attack medication online order panic buy propecia online how propecia works buy insurance medication without xanax casino eagle soaring valium sale casino gambling internet online cheap online order phentermine online casino game cheap phentermine no prescription needed buy buy target xanax xanax casino vegas propecia result casino deposit no propecia and woman buy link member spacesmsncom xanax casino slot online prescription propecia baldness hair lasercomb loss propecia casino news online casino commerce casino payout addicted to ativan casino hard rock seminole casino pechanga propecia sale propecia picture propecia cost buy xanax online pay by check casino fortunelounge online 2mg buy generic xanax buy xanax overnight propecia work casino online poker casino hotel paris propecia body building advice casino online cheap phentermine no prescription required propecia vs rogaine picture pill valium casino forum online ativan ativan buy cheap online valium versus xanax ativan benefit casino game online play cheap est phentermine casino directory gambling online casino internet what does valium do ativan and pregnancy 2005com buy link onlinebuy xanax casino download free online blogspirit cheap com online phentermine buy cheap link onlinecom xanax xanax does effects propecia side work capri casino isle casino gambling advanced casino conscious gambling investor online roulette system tax technique ultimate winning ativan effects hair loss propecia best price for propecia ativan combining neurontin tramadol get propecia cache casino creek order propecia cheap phentermine table casino guide online casino online review california casino casino free valium ativan casino poker buy online prescription propecia vaniqa viaga xenical propecia patent propecia effectiveness cheap phentermine online propecia rogaine propecia patent expiration casino casino casino online top casino winner propecia hair loss treatment generic propecia canada casino wager loss propecia casino las vegas best casino buy no online prescription xanax propecia alternative 4younarodru buy link onlinehtm xanax xanax casino free machine online slot breakthrough growth hair propecia archive blog buy inurl propecia approval cheap doctor phentermine without archive ativan blog buy inurl online casino chip poker cheap diet online phentermine pill casino motion original picture soundtrack discount propecia casino internet online casino gamble propecia forum casino keno online blog cheap phentermine casino online us casino rama benzodiazepine buy xanax cheap cheap phentermine casino reno finasteride propecia ambien sleep walking cheap phentermine c o d ativan doctor withdrawal casino free online buy xanax xr cheap danger menopause phentermine ativan from withdrawal casino online propecia baldness cheap phentermine cod casino online slot casino online uk casino foxwoods propecia testimonials online casino gambling barbary casino casino station affiliate casino online program cheap comment info personal phentermine remember propecia dosage does propecia work casino gambling internet cheap propecia cheap price on phentermine cheap com link online phentermine splinder buy xanax overnight shipping ativan dose effects side propecia without a prescription buy cheap cheap online xanax xanax atlantic casino city liquid propecia buy link onlinehtm pharmacyrxfreewebtoolscom xanax bonus casino online atlantic casino city hotel hair loss baldness propecia cheap perscriptions phentermine bonus casino free online propecia tab pay pal buy xanax buy generic propecia black casino jack casino online riverbelle bet casino online cheap phentermine no prescription free shipping propecia review adderall ativan cheap phentermine no rx phentermine cheap com cheap phentermine diet pill casino echeck online valium overdose buy 2mg xanax cheap phentermine overnight delivery propecia canada buy generic xanax adipex cheap phentermine valium stress test buy propecia buy xanax cash on delivery
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
The Test
Front Page
Do not turn this page until instructed.
Math 240 Algebra I - Term Test
University of Toronto, October 24, 2006
Solve the 5 problems on the other side of this page.
Each of the problems is worth 20 points.
You have an hour and 45 minutes.
Notes.
- No outside material other than stationary and a basic calculator is allowed.
- We will have an extra hour of class time in our regular class room on Thursday, replacing the first tutorial hour.
- The final exam date was posted by the faculty - it will take place on Wednesday December 13 from 2PM until 5PM at room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health).
Questions Page
Solve the following 5 problems. Each of the problems is worth 20 points. You have an hour and 45 minutes.
Problem 1. Let be a field with zero element , let be a vector space with zero element and let be some vector. Using only the axioms of fields and vector spaces, prove that .
Problem 2.
- In the field of complex numbers, compute
