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- Apr 14, 2013

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Let $1\leq n,k\in \mathbb{N}$ and let $v_1, \ldots , v_k\in \mathbb{R}^k$. Show that:

- Let $w\in \text{Lin}(v_1, \ldots , v_k)$. Then it holds that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(v_1, \ldots , v_k,w)$.
- Let $v_1, \ldots , v_k$ be linearly dependent. Thn there is a $1\leq i\leq k$ and $\lambda_1, \ldots , \lambda_k$ such that $v_i=\lambda_1v_1+\ldots +\lambda_{i-1}v_{i-1}+\lambda_{i+1}v_{i+1}+\ldots +\lambda_nk_n$.
- Let $i_1, \ldots i_k\in \mathbb{N}$, such that $\{1, \ldots , k\}=\{i_1, \ldots , i_k\}$. Then it holds that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(v_{i_1}, \ldots , v_{i_k})$.
- Let $v_1, \ldots , v_k$ be linearly dependent. Then there is a $1\leq i\leq k$ such that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(v_1, \ldots , v_{i-1}, v_{i+1}, \ldots, v_k)$.

I have already shown the first two points.

Could you please give me a hint fot the point $3$ ?

As for point $4$ : Do we use here the point $2$ ? Suppose $v_i=\lambda_1v_1 +\ldots \lambda_{i-1}v_{i-1}+\lambda_{i+1}v_{i+1}+\ldots +\lambda_kv_k$. Then it holds that $\text{Lin}(v_1, \ldots , v_k)\subseteq \text{Lin}(v_1, \ldots , v_{i-1}, v_{i+1}, \ldots, v_k)$, or not?

No it is left to show that $\text{Lin}(v_1, \ldots , v_{i-1}, v_{i+1}, \ldots, v_k)\subset \text{Lin}(v_1, \ldots , v_k)$, or not?

Or is there an other for this proof?