Linear Algebra Math 221
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1 Linea Algeba Math Open Book Eam Open Notes Sept Calculatos Pemitted Sho all ok (ecept #). ( pts) Gien the sstem of equations a) ( pts) Epess this sstem as an augmented mati. b) ( pts) Bing this mati to educed echelon fom. c) ( pts) Is the sstem consistent? If so ite don all solutions of the oiginal sstem. YES The sstem is consistent as the educed echelon fom (the echelon fom ould be adequate) has no os of the fom ( ) in othe ods no equations of the fom. We note that the fouth column the column has no piot so is a fee aiable. We no ead off the est of the solution fom the os of the educed echelon fom. The fist o eads ()()()(-)- so - the second o eads ()()()(-)- so - and the thid o eads ()()()(-)- so -.
2 is fee Check: We check b plugging back into the oiginal equations: - (-)-(-)(-) - (-)-(-)() - (-)-(-)(-)(). ( pts) Gien A a) ( pts) Bing mati A to echelon fom. b) ( pts) Which columns ae piot columns? Columns and ae piot columns. (Column is non-piot.) c) ( pts) Is the sstem A consistent? YES An homogeneous sstem is consistent. As the augmentation column (hich e ael bothe to ite don fo a homogeneous sstem) contains onl eos e cannot possibl get an inconsistent o of the fom ( ). d) ( pts) Continue fom echelon fom (pat a) and bing the mati to educed echelon fom. e) ( pts) Wite don all solution ectos to the sstem A. Fom the educed echelon fom poduced in (pat d) e ead off the anse. The onl non-piot column is the thid so is a fee aiable. The last o
3 gies us an equation ith no content. The fist o gies us the equation () () (-) and the second o gies us the equation () () (). Thus the solutions ae:{ - is fee} and putting the solution in ecto fom e get: o an multiple of the ecto ( - ) Check: We check b multipling A. ( pts) Conside the set of ectos? a) ( pts) Is the set { } lineal independent? NO In R (the space of -long ectos) no moe than ectos can possibl be lineal independent. b) ( pts) Find a linea dependenc beteen and if one eists. We ant to find a set of alues fo {a a a a } not all eo such that e get a linea dependenc a a a a. We sole this homogeneous equation foming a mati and pefoming o eduction: As the thid column is non-piot a is a fee aiable and e ead off the equations fo the aious os: ()a ()a ( -½ )a ()a so a a / ()a ()a ( ½ )a ()a so a -a / ()a ()a ()a ()a so a If all e need is a single dependenc set a to some non-eo alue sa a so {a a a a } { - } and e get the dependenc: a a a Check: We confim b plugging back in:
4 a a a a () ( ) () () c) ( pts) Is in the span of { }? If so ite as a linea combination of the othe thee ectos. YES As is used in the dependenc e can eite it as: - d) ( pts) Is in the span of { }? If so ite as a linea combination of the othe thee ectos. NO As is not used in an dependenc e cannot eite in tems of the othe ectos.. ( pts) ALWAYS Tue NEVER Tue o SOMETIMES Tue ( pts pe question) a) Matices hich ae o equialent hae the same educed echelon fom. ALWAYS Tue Each mati (and class of o equialent matices) has eactl one equialent educed echelon mati. (see La d Ed Chap : Thm and Supplementa Poblem #(i)). b) A linea tansfomation fom R to R is -to-. NEVER Tue Recall that such a linea tansfom is epesented b a mati ( os and columns). When ou educe such a mati ou can hae at most piots and thee ill cetainl be non-piot columns. Recall that being -to- is equialent to haing no non-piot columns. Also ecall that haing non-piot columns means that ou hae fee aiables so that if the equation Ab is consistent it has man solutions not the unique solution of a -to- tansfom. c) A linea sstem of equations ith aiables is consistent. SOMETIMES Tue An obious eample is a homogeneous sstem A hich is alas consistent. Conesel if the sstem had an equation of the fom (o such an equation as poduced b o eduction) then the sstem is not consistent. Note that this agument hold tue fo an numbe of equations in an numbe of aiables.
5 d) A homogeneous linea sstem of equations ith aiables has a unique solution. NEVER Tue While it is tue that a homogeneous sstem (i.e. A) alas has at least one solution - the tiial solution if thee ae moe aiables than equations ou ae guaanteed to hae moe than that solution. Note that the coesponding mati A ill hae thee os and columns. As it can hae no moe than piots at least one of the columns ill be non-piot and thee ill be a fee aiable. e) Matices ith the same echelon fom ae o equialent. ALWAYS Tue A mati is o equialent to its echelon fom (this is hat e mean b saing its echelon fom) and thee is a sequence of o opeations coneting that mati to its echelon fom. So e hae to matices each equialent b a knon sequence of o opeations to a common echelon fom. We no poduce a sequence of o opeations taking us fom one mati to the echelon fom to the othe mati. This sequence shos that the to matices ae o equialent.
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