Prove that if vectors $v_1,v_2...,v_k$ in a vector space V are linearly independent and a vector $u\in V$ is such that $u\notin \text{lin}(\{v_1,v_2,...,v_k\})$,then the vectors $v_1,v_2,...,v_k,u$ are linearly independent.
My attempt:
$u\notin \text{lin}(\{v_1,v_2,...,v_k\})$ since clearly $u=0\cdot v_1...+0\cdot v_k+1\cdot u$ so $0\cdot v_1+...0\cdot v_k+u(1-1)=0$ , and given that $v_1,...v_k$ are linearly independent implies $\text{span}(\{v_1,...v_k,u\})=0$ if only if all real coefficient $=0$.
I guess my answer is wrong, any help , thanks.
Here's a good way to think about it.
Suppose The set $\{ v_1, \ldots, v_n, u\}$ form a linearly dependent set. Then one is a linear combination of the others - say,
$$ u = \sum_{j = 1}^n c_j v_j $$
and so $u \in \operatorname{Span} \{v_1, \ldots, v_n \}$.
Thus the original statement holds.