Question

1 2 -1 0 0 1 0 0 -1 3 ſi 2 0 2 5 [10 (11 points) The matrix A= 2 1 3 2 7 reduces to R= 0 3 1 a 6 5 0 1 Let ui, , 13, 144, and us be the columns of U. (a) Determine, with justification, whether each of the following sets is linearly independent or linearly dependent. i. {u1, 12, 13) ii. {u1, 13, us} iii. {u2, 13} iv. {u1, 12, 13, 1} (b) State the span of each set of vectors in part (a)(this should have 4 answers; one for each set) (c) List the sets in part (a) that are a basis for R3. Explain your answer using the definition of a basis.

Answer 1

Ilin early dependendent ( ح ۱۰ ( 1 , 2 , e ( wے 0 2 2 det - - ) و s Expanding Wirto lo Colum-1] وه ر دلار بوده که به او می ate ans. aci) 2 / and در) 2 we find find that * Us = 20 - U2 (2)-(2) 1/) دور من , و : هه ر وا } هم د 2 و as b(7) & a) {a ( ا W 3, us

ded (us U ,U3, 45) 2 U 3 - 2 19 -1 3 0 0 Column-1) Wor, to ( Expanding - 117 3 -3 th 0 independent atce ans of a (W) Hence { 1, 142, 43 linearly {4,4,43} 2 span qui U, 42, 43 L: اروا ر ans. 6()

. a cel {ua, uz? zig (3), (3) q Up+(₂ U₂=0 ♡ () (3)-(0) ().(:) >> 26 {242 { C1-C2=0 92(2=0 are Hence Į U2, U3} linearly independent ans. 4 acie and { 42, 43] Span & usus as. fbce) linearly acil {4,342 9 U3, U43 ? {41242243}; a dependent sex [ from paret acom] is I linearly Q Hence { ",,U2, U3, U4 dependent set. dont on the ans, of aciv)

since (:)-(; )+ (3) 3 au, -u2+3 43 { up, U₂, U3} { 0, 49, 439 44} - Span sosy ans. of brin Look carefully Redmark b). (h part (a) Answers polés. here there is only one c) Jets in part can Out of Three basis, which is {41.393,4s} . linearly and assuz) and Since eCA) 8 is a subspace of R 3 independent lis span of us uz, us] CCA) by defination of basis, {1,2 43 44} form the and hence by basis set.

d) Yur, U2, U3} sie linearly dependent, linearly dependendo of A Cor! -) and Col.3 Coh 2 2 0 2) 2 3 3 -> 2 0 -3 3 R₂ R2-281 R3 R3-3R, -5 c 2) -3a + 15 = 0 a=5 2 5

e) Led Ax: 0 =Y RX=0 2. 42 죠 0 0 0 12, 게 0 ( 게5 IN 0 lx sut 3 f 0 6 42 - 13 Su- 0 키 54 거서 sut -나 +73 jg > 지 =0 7ding) 4a2=0 2=-1) Subtractis 3 13 지수의 경 - من و ت +24 드 2 -3. 다. 다. üns 25 X2 8 - 이 3 2 Ks 2 - 14. hy 지 TW 73 지 4 4 HW Su 74 x 지 수 TO00

independent and c linearly auce . Nul(A). and generates Nul(A) is a basis of 2 and Nallity of A and and, Nul (A) » Spom -2012
Note: rest part of d given below: evaluation of missing value of b:

32 - My 1 2 25 27 १।। 2 3 3 01 Į R2 ८ 25 2 wo 6-1 -10 0 25 2. P4- 26L R₂ +5R2 - . 01-7-१ 3 5 - - 2 3 3 2 O । -1 3 -5 C-5 S: (:0-50 2 D 7 wr O mian 6 . 0 ६ 2 - F2-29 8 3 " 0 2 - - + 21, 1 ० । -36 G 0 ४ and asb (८ 2. ६ 37 21-2- १) - +९ - - -8 ३) :) . 1: ।

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