exam1sol - Marl'h HO- T—EJL 219%“HORM l. (5 points...

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Unformatted text preview: Marl'h HO- T—EJL 219%“HORM l. (5 points each) Determine whether each of the following assertions is true or false. Give a brief explanation for each answer (full proof is not required). (a) If a linear transformation T: V —> W between finite—dimensional vector spaces is 1—to—1, thendim<gzuidifi3?. (T3110’ dimLV3=muT) ‘4'} l“ Ohmgw lzw‘e°\' Turn-L. Awa. rmKCT)‘-_J-lmu)) Shine Tact) €vJ. (b) If V and W are finite-dimensional vector spaces such that dim(V) S dim(W), and T: V ——> W is a linear transformation, then T is 1—to—1. Elm. A cka is Tm. am we»? T: tRn'MR“ #1? “*7 ">0. (0) The set of vectors ($1, $2, $3, .724) which satisfy x1 = 3:4 and 232 = x3 is a subspace of R4. TM. TN Slv-«YVS'l’ "mum‘s M part (cl) is we M. (d) The set of vectors in part (c) is the nullspace of a linear transformation from R4 to some vector space over R. Tm. m m wufiyau a; $11244”? (£th b3 “Bowman 2 (X.'X«+,‘K7:Y3). (14'; «.651 in check {ML T rs “(CW bwl yum will Widow it get Hi we“ w. M [701‘th) (e) The set of vectors in part (c) is the nullspace of a linear transformation from R4 to R (in other words, a linear functional). ma. 3m T a PM (d) is macho, 4 we kaT)=Z, we WM “'8 mllscu-L, £2ng ‘5 “M 9:) r4 veal-w; {of Lt), Lu; 091%wa Z, Edi “In mlbfau 4 M3 Jamar Saw-*le lMd ohmsm 22 by fimgm M . (f) Q” is a subspace of the vector space R” over R. (Q denotes the field of rational numbers.) Elsa. Nol acme «Mair smflax waUCwl-I‘un by ivmla‘wl mph/3 (g) Q” is a subsagce of R” considered as a vector space over Q (with the usual addition, and multiplication by rational scalars). Tml . Cuwifi chunk WW «0%le amQ clamp WAD»! ;cc.lar W“ “Wilhelm S‘w- CW” ~- WM) 6 ((2K Ii a. M «M X; m V‘otl'imxcl . 2. Let S be the following subset of P(R): S = W) = x5 + x2, gm = x5 + 2, he) = x3, M) = as? — 2} (a) (30 points) Find a subset of S which is a basis of Span(S) and prove that your answer is correct. Tmm ML M \nSSWLa Wd' Msm: 4M7 9.4m)— my‘sh“? v4. 2‘ MR {we 2M QM 3.00;), 0:), ‘(m§_ Tu Fvw-e to S J M M Mr lanes B:%-C(m)’%(x)’\obt3§ :5 a was. Tm PM o guitar-Sm S?M(%):S\0M(S). 9mm BC—S, $pm(¢$)¢—Spm(£>, N42 J4, ere 9pm53999mU5), 93mm S‘3M(\3) is a Su‘JS act, H's euomgk 40 FM SESpchs). Twas we Duh, MQoQ k ska» M4 Jon :g m 89mm), cath ‘m M Lxcausa jLX) = {300—500 . \ la, Nuu wg’u gm 8. is 9AM SWDSQ a, c (pg-L S W SK»ka i‘ av A£(x)-\-\93Lx) +C.\A.(.X) “0 0:,ka a} yobijcis), Tm £41ko “an is (M5316; +c.x3 +a$7‘ 1"Uo . Fm” Ms 4-» he M D Pot3wc£ ova m5“! Law a=E=C°O, (b) (5 points) Find dim(Span(S)). amuvmm = \fb\ =3. 3. (30 points) Let T: V ——> W be a linear transformation. Prove that if T is l—to—1, and v1, . . . ,Uk 6 V are linearly independent, then T(vl), . . . ,T(vk) are linearly independent. Sammy“ Q\T(\J\) + QZTLVL3+ + ak T(\Ju\ =0 , 9mm 1‘ a: 0m, +142 RAH-howl) 5:00,). is naval i-o TLay‘ +azv,_«-... +qumy S‘mm T is l— {‘04, ’iC-JL ‘Cad- ‘lbq‘l' {his Em («fix-25 Q‘m + QzVL-t- . . -+C\\‘V\L:o. ' ~ J. a; fine/UL}, QMUL UH ...l\jv_ m wilsz , S \m‘nbu Mix all coaCL‘cJeMls a; m 200. “M62 TWO ) “VT-(Va) CUM flaws \uaflnPQMQ“-+- ...
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exam1sol - Marl'h HO- T—EJL 219%“HORM l. (5 points...

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