Question

2. Let V2 = R', the set of all 3-D vectors, with vector addition and scalar multipli- cation defined as follows: • if a = (a1, 02, 03) and b = (b.b2, 53) are vectors in V2, then a + b = (a + b +5,22 + b2 - 7. a3 + b3 + 1) • if n E R and a = (01, 22, 23) EV, then na = n(a), 22, 23) = (naj +5(n-1), na2 - 7(n-1), nag+n-1) Determine whether V, under the operations of vector addition and scalar multi- plication defined above forms a vector space. If so, explain why. If not indicate at least one vector space axiom that fails to hold for V2.

please solve using all 10 listed bellow:
1. closure property of addition,
2. commutative property,
3. associative property,
4. additive identity property,
5. additive inverse property,
6. closure property of scaler multipication,
7. vector distributive property
8. scaler distributive property,
9. scaler associative property
10. scaler identity property

Answer 1

7 Let -(4,,)md T-(, ba) Ahen +(4th+, 4+h-2, tbg+) E V clorume prerty holdo New, +(4+b+5, 4th,Astboti) (b1ta+5, byta,, bstt =+ fer a , n Afonanucatie pdorty all Lt -(a,4,4), ( b),-,,) Phen C+ 4+b+, 4th,4+bo++ Attttt = (4tb+4+10, 4tbyt 14,4thtete) +)(t4+5, byt, bot+) -(4tb+9+5b+&- t, 4thst+ (4tbytt10, tt14,tbt Cot (+ (E4)r aa,, in Aociatie prerty hdldo Hore Es,,-i) the alitive identty Here ED-A, 14-,-- he inverst f in 4,4, Afor al

Lot nER nd -(a,4,) Mhen nn(4,4,44+s-),mat), nag+n-)E tor all mER and eV Clauore mfurty hida n (a+t)=n(4thtr, ath, tht = (n(ath+)+5-),n (4th-2), n(4tbott -i) (matnhytstsm-g ntnb,n-nt man tnbs+ntn) n4nbyon-5 natnbs 4nt, mannA+ , ma-), nagt n- + Cnby+s(-), mby (1-), mbotn-s Cn4tnyton-0, ng+nb-14nt14, matnbgten-e al nER nd R ,t EV Disbribucive puferty oa nt hld .daa nr fam . Vaeton