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- Apr 14, 2013

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Hey!!

Let $1\leq k,m,n\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $U$ a subspace of $V$ with $\dim_{\mathbb{R}}U=m$. Let $u_1, \ldots , u_k\in U$ be linear independent. Show that there are vectors $u_{k+1}, \ldots , u_m\in U$ such that $(u_1, \ldots , u_m)$ is a basis of $U$.

Hint: Use the following two statements. Convince that $\ell:=m-k\geq 0$ and use induction on $\ell$.

Let $1\leq k\in \mathbb{N}$ and $v_1, \ldots v_k, w\in V$.

Let $U\leq_{R}V$ and $W\leq_{R}V$ such that $W\subseteq U$. Then it holds that $\dim_RW\leq \dim_RU$ and the equality iff $W=U$.

So to use Statement 1 we have to show that $u_{k+1}, \ldots , u_m\notin \text{Lin}(u_1, \ldots , u_k)$ and then we apply the Statement 1 using induction or how do we have to start?

Let $1\leq k,m,n\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $U$ a subspace of $V$ with $\dim_{\mathbb{R}}U=m$. Let $u_1, \ldots , u_k\in U$ be linear independent. Show that there are vectors $u_{k+1}, \ldots , u_m\in U$ such that $(u_1, \ldots , u_m)$ is a basis of $U$.

Hint: Use the following two statements. Convince that $\ell:=m-k\geq 0$ and use induction on $\ell$.

**Statement 1:**Let $1\leq k\in \mathbb{N}$ and $v_1, \ldots v_k, w\in V$.

- $1\leq m\in \mathbb{N}$ and $w_1, \ldots , w_m\in V$ such that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(w_1, \ldots , w_m)$ then it holds that $\text{Lin}(v_1, \ldots , v_k, w)=\text{Lin}(w_1, \ldots , w_m, w)$.
- If the vectors $v_1, \ldots , v_k$ are linear independent and $w\notin \text{Lin}(v_1, \ldots , v_k)$ then also the vectors $v_1, \ldots , v_k, w$ are linear independent.

**Statement 2:**Let $U\leq_{R}V$ and $W\leq_{R}V$ such that $W\subseteq U$. Then it holds that $\dim_RW\leq \dim_RU$ and the equality iff $W=U$.

So to use Statement 1 we have to show that $u_{k+1}, \ldots , u_m\notin \text{Lin}(u_1, \ldots , u_k)$ and then we apply the Statement 1 using induction or how do we have to start?

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