Suppose that V is a vector space, that $\displaystyle X=\{v_1,...v_n\} \subset V$ and that $\displaystyle w \in span X$. Set $\displaystyle Y=\{w, v_1,...,v_n\}$. If necessary you may denote the field of scalars by F.

**(a)** Show that $\displaystyle Y$ is linearly dependent.

**(b)** Show that $\displaystyle spanY = spanX$

This is my attempt for part (a): We know that since w is in the span of X then it's a linear combination of its vectors (i.e. $\displaystyle w=a_1 v_1 +...+a_n v_n$, and $\displaystyle a_1,...,a_n$ are scalars).

Suppose there are real numbers $\displaystyle \lambda_1, \lambda_2,...,\lambda_m$ (m=n+1) such that:

$\displaystyle \lambda_1 w+ \lambda_2 v_1 + \lambda_m v_n = 0$

Then $\displaystyle Y$ is linearly dependent if $\displaystyle \lambda_1 = \lambda_2 =...= \lambda_m = 0$. Now I'm not sure how to prove this.

Of course we can rewrite it as:

$\displaystyle \lambda_1 (a_1 v_1 +...+a_n v_n)+ \lambda_2 v_1 + \lambda_m v_n = 0$

But that doesn't seem to help...