- #1

mathmari

Gold Member

MHB

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Let $1\leq n\in \mathbb{N}$ and $v_1, \ldots , v_k\in \mathbb{R}^n$. Show that there exist $w_1, \ldots , w_m\in \{v_1, \ldots , v_k\}$ such that $(w_1, \ldots , w_m)$ is a basis of $\text{Lin}(v_1, \ldots , v_k)$.

I have done the following:

A basis of $\text{Lin}(v_1, \ldots , v_k)$ is a linearly independent set of vectors of $\{v_1, \ldots , v_k\}$.

So let $\{w_1, \ldots , w_m\}\subseteq \{v_1, \ldots , v_k\}$ be a linearly independent set.

$\text{Lin}(v_1, \ldots , v_k)$ is the set of all linear combinations of $v_1, \ldots , v_k$. So it left to show that we can express every linear combination of that set using the vectors $\{w_1, \ldots , w_m\}$, or not? (Wondering)