Question

Let S = {α, β, γ}, T = {α, α + β, α + β + γ}, W = {α + β, β + γ, α + γ} be subsets in (4)

a vector space V. Prove that L(S ) = L(T) = L(W).

Answer 1

$s=(\alpha,\beta,\gamma)$

let $t \in L(T)$ where t is a linear combination of element of T.

i.e. t=$a(\alpha)+b(\alpha+\beta)+c(\alpha+\beta+\gamma)$

so $t\in$ L(S)

L(T)$\subset$ L(S).................................(1)

Now, w$\in$ L(W) such that $w=a_1(\alpha+\beta)+b_1(\beta+\gamma)+c_1(\alpha+\gamma)$

w=(a₁+c₁)$\alpha$+

(a₁+b₁) $\beta$ +(b₁+c₁₎$\gamma$

w is a linear combination of $\alpha,\beta,\gamma$

so w$\in$ L(S)................................(2)

now let s$\in$ L(S), where $s=a_2+b_2\beta +c_3\gamma$

$s=a_1\alpha+b_2\beta+c_3\gamma$

we can write s in form of

$s=a_3\alpha +a_4(\alpha+\beta)+a_5(\gamma+\alpha)$

and

$s=a_6 (\alpha+\beta)+a_7(\beta+\gamma)+a_8(\gamma+\alpha)$

so, s$\in$ L(T); s$\in$ L(W)

$\therefore L(S)\subset L(T); L(S)\subset L(W)$

$\therefore L(S)=L(T)=L(W)$

hence proved...