If {$B\vec{v}_1,...,B\vec{v}_k$} is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k$ x $n$ matrix, ...

Question

If {$B\vec{v}_1,...,B\vec{v}_k$} is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k$ x $n$ matrix, then {$\vec{v}_1,...,\vec{v}_k$} is a linearly independent set in $\mathbb{R}^k$.

I need to either prove or disprove the following statement. How do I start?

Answer 1

Hint: Use the definition of linear independence. Assume that $a_1 \vec{v_1} + \ldots + a_k \vec{v_k} = \vec{0}$, multiply this equation on the left by $B$ and see what you get.

Answer 2

Assume $\{\vec v_1,\dots,\vec v_k\}$ is linearily dependent then you have some coefficients $c_2,\dots,c_k$ with

$$ \vec v_1 = \sum_{j=2}^k c_k \vec v_k $$

applying $B$ on both sides yields

$$ B\vec v_1 = \sum_{j=2}^k c_k B\vec v_k. $$

This means $\{B\vec v_1,\dots,B\vec v_k\}$ must have been linearily dependent, which is a contradiction. Thus $\{\vec v_1,\dots,\vec v_k\}$ has to be linearily independent.