Jtdylktuy

Consider a vector space V , vectors v1,...,vn ∈ V and

w = $α_1v_1$ + $α_2v_2$ +···+ $α_nv_n$, where $α_1$,...,$α_n$ ∈R and $α_1$ $neq$ 0.

Prove that:
a) If $v_1$,$v_2$,...,$v_n$ are linearly independent, then w,$v_2$,...,$v_n$ are linearly independent

How would I a formulate proof for this or even the starting step?

HINT

Let apply the definition to the set $w,v_2,ldots,v_n$ to obtain

$$c_1w+c_2v_2+ldots+c_nv_n=c_1(α_1v_1 + α_2v_2 +cdots+ α_nv_n)+c_2v_2+ldots+c_nv_n=ldots=0$$

Can you continue form here? What condition do we find for $c_1,c_2,ldots,c_n$.

Kickstarter. Consider a linear combination $0 = c_1 w + c_2 v_2 + ldots + c_n v_n = d_1 v_1 + d_2 v_2 + ldots + d_n v_n,$ where the $d_j$ are explicitly known from the $c_j$ (you do it). Conclude each $d_j = 0$ and then each $c_j = 0.$

Start with the definition of what you are trying to prove. You want to show that if $c_1 w + c_2 v_2 + ... +c_n v_n = 0$, then all the coefficients $c_i$ must be 0. It might be easier to try proving the contrapositive of the statement. Assume that there exist $c_i$ that are not 0 and show that $v_1,...,v_n$ are linearly dependent.

Suppose $a_1w+a_2v_2+cdots + a_nv_n = 0$. If $a_1 neq 0$, then $alpha_1v_1+alpha_2v_2+cdots + alpha_nv_n = w = -dfrac{a_2}{a_1}v_2 - dfrac{a_3}{a_1}v_3 - cdots - dfrac{a_n}{a_1}v_nimplies alpha_1v_1+displaystyle sum_{k=2}^n left(alpha_k+dfrac{a_k}{a_1}right)v_k = 0implies v_1,v_2,..., v_n$ are linearly dependent, since $alpha_1 neq 0$ - a contradiction. Thus $a_1 = 0$ and since $v_2,v_3,...,v_n$ are linearly independent, we have $a_2=a_3=...=a_n = 0$, proving that $w, v_2,v_3,...,v_n$ are linearly independent.