OK, here's another lemma I'm being asked to prove and I am trying to see if I am in the right ballpark.
Let $L$ = {$y, y_1, y_2, y_3,…, y_m$} be a linearly independent subset of $V$, a vector space over field $K$.
Let $S$ = {$x_1, x_2,…,x_r$} be a subset of $V$ such that
{$y_1, y_2,…,y_m, x_1, x_2, x_3,…,x_r$} = ($L$ – {$y$}) $\cup\ S$ spans $V$.
Show $\exists$ an element $x_i\in\ S$ s.t. $L \cup\ $($S$ – {$x_i$}) = {$y, y_1, y_2,…, y_m, x_1, x_2, x_3, …,x_{i-1}, x_i, x_{i+1}, x_n$} spans $V$.
I approached it like this:
Given that ($L – ${$y$}) $\cup\ $ $S$ spans $V$ , that means that $y$ is linearly dependent, and thus a linear combination of $L \cup\ S$, so it can be expressed:
$y = \lambda_1y_1 + \lambda_2y_2 + … \lambda_m y_m + \mu_1x_1 + \mu_2x_2+ … \mu_rx_r$
by construction L is a linearly independent set, $\exists$ scalars $c$ such that $cy + c_1y_1 + c_2y_2 + … c_my_m = 0 $
But since we have L as linearly independent, that means the coefficients have to be zero. That's true for both L as a linearly independent set with scalar coefficient $c$ and $\lambda_i$.
That in turn reduces $y = \lambda_1y_1 + \lambda_2y_2 + … \lambda_m y_m + \mu_1x_1 + \mu_2x_2+ … \mu_rx_r$ to $y = \mu_1x_1 + \mu_2x_2+ … \mu_rx_r$
But y isn't linearly independent. That means at least one of the coefficients $\mu_i \neq 0$ therefore there is some element $x_i \neq 0$
The span of V is the set of all vectors in V. And we're trying to show that the union of L and S with that nonzero $x_i$ subtracted is the span of V. But that's where I get a little stuck; I know I've almost got the last step.
(also, any critique o how to write this proof "properly" is much appreciated).
Best Answer
You don't know that $\{ y, x_1, \dots, x_r \}$ is a linearly dependent subset! You are misunderstanding the definitions. The problem is that when you write $$ y = \lambda_1 y_1 + \dots + \lambda_m y_m + \mu_1 x_1 + \dots + \mu_r x_r, $$ you cannot assume that all coefficients $\lambda$ will be zero. Think of $V = \mathbb R^2$ with basis $\{e_1,e_2\}$, $L = \{ e_1, e_2 \}$ and $S = \{e_1 + e_2\}$ and see what happens with your equations.
The right proof would go like this : since $\{ y_1,\dots,y_m , x_1, \dots, x_r\}$ spans $V$, we have $$ y = \lambda_1 y_1 + \dots + \lambda_m y_m + \mu_1 x_1 + \dots + \mu_r x_r, $$ and we cannot have all $\mu_i$'s equal to zero, because this would give a linear dependence on the set $\{y,y_1,\dots, y_m\}$. Therefore one of the $\mu$'s is not zero. Say $\mu_i \neq 0$. Then $$ x_i = \frac {-1}{\mu_i}\left( \lambda_1 y_1 + \dots + \lambda_m y_m + \mu_1 x_1 + \dots + 0 x_i + \dots + \mu_r x_r - y \right), $$ which means even though the set $\{y,y_1, \dots, y_m, x_1, \dots, x_r \}$ spans $V$, the element $x_i$ is not necessary to span $V$, so we can remove it from this set and still have a generating set for $V$.
Hope that helps,