Question Number 148222 by BHOOPENDRA last updated on 26/Jul/21

Answered by iloveisrael last updated on 26/Jul/21

v_3 =λv_1 +αv_2 (((−3)),(( 4)),(( 7)) ) = ((( λ)),(( 0)),((−λ)) ) + ((( 0)),((2α)),((2α)) ) { ((λ=−3)),((α=2)) :}⇒v_3 =−3v_1 +2v_2

Commented byBHOOPENDRA last updated on 26/Jul/21

what about the other two sir ?

Answered by Olaf_Thorendsen last updated on 26/Jul/21

(ii) Let v = ((x),(y),(z) ) ∈ W then ∃(α,β,γ)∈R^3 \ v = αv_1 +αv_2 +γv_3 v = αv_1 +βv_2 +γ(−3v_1 +2v_2 ) v = (α−3γ)v_1 +(β+2γ)v_2 Let a = α−3γ and b = β+2γ then ∃(a,b)∈R^2 \ v = av_1 +bv_2 That proves that (v_1 ,v_2 ) generates W. (iii) We solve the equation λv_1 +μv_2 = 0_W (1) ⇔ λ ((1),(0),((−1)) )+μ ((0),(2),(2) ) = ((0),(0),(0) ) ⇔ (λ,μ) = (0,0) We canno′t find (λ,μ)∈R^2 −{(0,0)} such that (1) is true. That proves that v_1 and v_2 are linearly independent.