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Lecture13and14filled

# Lecture13and14filled - University of Toronto at Scarborough...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer 85 Mathematical Sciences MAT A23 Winter 2008 Lecture 13 and 14 Sections 2.2, 2.3 Sophie Chrysostomou 11212 2m1111 EXAMPLEO.1.A—— 1 2 3 1 3 21333 The T0?!) space of A is the span of the row vetors of A, therefore the row Space of A is given by the set ’R, ' R : 51061:1:2a132]v[22”13111a1]a[1:2w33113}7[2211333?3]) = {7‘1[1,1,2,1,2]+T2[2,—1,1,1,1]+T3[1,2,3,1,3]+T4I2,1,3,3,33 T1,?2,T’3,T’4ER} 7?, is a subspace of R5. The column space of A is the span of the column vectors of A, therefore the coiumn space of A is given by the set C 1 1 2 1 2 2 —1 1 l 1 C " 5p I a 2 F 3 a l 7 3 1 2 1 3 3 3 1 1 2 1 2 —— 1 I = 01 i +612 ; +513 33 +a4 1 +a5 3 €11aaeaﬂeﬂ4aﬂ5€R 2 1 3 3 3 C is a subspace of R4. How to ﬁnd bases for 7?, and C? THEOREM 0.2. Let A be row equivalent to H Where H is in a REF or a REEF, then a) If the the column space of A is C , then a basis for C, consists only of ail the columns of A, corresponding to the pivotal coiumns of H. b) if the row space of A is R, then the senzero rows of H form a basis for R. c)dim(C)ﬂdim(R). their bases have. ”the same. nowhere o‘F vest/0W d) If the nuilspace of A is N, then dimUV) m the number of nonpivotal columns of H. PROOF. (of (b) and (c) only ‘0) ANH H is in REFW REEF “the mom-2.9m vows 0’? H one [in maize. ‘ We need to show &hm the span 0? honvZon rows 0‘3 H l\$ R_ This wiH show that «the mona'wm rows 0? H cm a, basis. WR_ Eve {\Ohw’lxxo vow 0&5 H is a wsuh a? a SQQMGMLQ. 9F e1emzntag row 0 mos done on matrices starting +‘ww(mw egmwalewt to) A. 6-663 nowwo vow 0H4 is o. lemr towbmmion o-F vows a? A . span (News 0’? H) C. SFCYOWS 0*: Pt) “On/Lem vow sfoce a? HER. QB Converse/13, H~I-\ ”‘Eveg NW 04: A is a. “wear WWbiha—W 0F nonpmro vows o-F H. 5‘ch rowsmm o—F H, @ _ (”‘90“0‘ 3M2. vowspacc o-F H ==‘ «R, ’“Spom UF nonv'zevo rows 0-? H :5 yv ”wt‘vuc vxovw‘lzvo vows 04: H we. at basis ~FoY (R. CJV‘mve. diwcyxﬁohw CC). . VWF: dwCC): it ch wluwns is A wrresfohdimﬂ «to piwm! bah/man m H. 1:11? 0F Fiwtod vavan» in H 596‘" OF nonwwro WWSQO‘F H. 7* OthJ’O 0‘) Prove ohm (X): #0": non piW‘ ooh/«Mans OF H‘ we: N: {3H Wnﬁ‘kfﬂ Hf???) S H has» K “OH” EW’CG Wluwng . __..~,.\ 23:: are, K 43%;; vgviaﬂes in «the ﬂawextﬂ SONUW “to A?" “*0 ' ﬁne gemevm solvrf/iOH to A‘X‘z'b" can be wvi-cten 0:9 0 [meow wwbs’wo‘ﬁm OFK \imarvg Kudzpenclzwt vac-bars Thrase, K vecbors owe. (a Mai; ~R>v X. " diwCJYh‘i—K. EXAMPLE 0.3. Find a basis for the column space, a basis for the row space, I I 2 1 2 a basis for the nullspace of A = A,._,_E*?&ip:::: 5H ooo®o 00000 RV\EF A bums ~R7r ﬁne, vowspom. 5 {E i. 0, I, 0J1, [0,! 1 011, [0,02021201} I)» A bask. £71m Muvwn Sfam\$i[é] [an] [t k I '1, t Z ’ l ’ 7: dim (vowsmm 0F Mzazacmcwmw SW 04? m 1 2 m1 1 1 1 1 2 3 l 3 2 I 3 3 3 >41, 39' a. m variable, X3==S x5 is on Hex, variable, Xs=t «sw-L “I w: -—*,. — «at. .,. «I "‘ X“ g “' i ”H3 0 o o o :..-.{[\,27,2, 0,1, [1,1,5,011,[2,b,5,81, DEFINETION 0.4. If A is an m x n matrix, then i) the dimension of the column space of A (which is the same as the di» mension of the row space of A) is called the rank of A, denoted by mnM/l). ii) the dimension of the nullspace of A is called the nullity of A denoted by nullity(A). THEOREM 0.5. The Bank Equation of (1 Matrix. Let A be an m X n matrix, row equivalent to H, where H is in REF or in REEF, then ranM/l) + nullity(A) = n Vireo-l? :lﬁrvjf ‘3in wWWms OFH ‘(OkaQQ 4r??? O'FMOYI’EthO'l Wluwms o‘F H “l" hmlll‘ﬁjCA3 52W o¥mlwlvwnso¥H a“ M EXAMPLE 0.6. Find a basis for the column space, a basis for the row space, I 2 0 2 1 0 a basis for the nullspaee ofAz 3 g i g _i _: and verify the rank 4 9 1 8 W1 ml equation. Y N 110150 io-wziogﬂg 7771612‘NUI10L1.’ WEI- 2\$i§v~\"\ OOOIOJZ’HW‘?‘ bfqlzl'l‘4 Ooooll ll [VIP E 1 3;; i .. ‘7 1 Easis 437v wlmvwn Seam o-F Ami 2. S 5 w; } El; liwliﬂ l3}. .. ., -' 01,IU,0 ’1, V79i\$l\$ \$0! WW SFGCX‘UF A»{4EI,O, 70020) \$1,]: 7:1; 14 [0,o,o,|,o,a2],[o,o,o,o,i,i .3 dimhow sFeuLCADT-Al: ohm ( oowvwn spacecm). Yank (Mall Z NuHsFaceLP‘): nuHsyocecﬁ): {T l A‘x‘wﬁ’}:{1‘|H-x’s‘o*3. XS+XBW W‘VWV‘V‘ 77, 17‘“ H bower If") P‘WC‘; 5X», Xb We 43% Vmaww'CQ/rs. X3212 X553. ___, Xi 55+2t t 1‘ +5 3 X X1- "? { O M- 18 0 X5 ,3 o ‘4 Yb S D l Homework: Find a basis for the column space, a basis for the row space, a 1 2 O 2 5 w? —5 1 ——1 ——8 basis for the nulkspace of A m 0 __3 3 4 I and verify the rank 3 6 0 _7 2 2 3 1 —6 ~l equation. 01 ...
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Lecture13and14filled - University of Toronto at Scarborough...

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