- If the vectors $a_1,...,a_k$ generate the subspace $V$ of $\mathbb{R}^n$, and $x\in\mathbb{R}^n$ is orthogonal to each of these vectors, show that $x\in V^\perp$.
My attempt: Let $y\in V$. Then, we can write
$$y=\beta_1 a_1+...+\beta_k a_k$$
for some $\beta_1,...,\beta_k\in\mathbb{R}$.
Since $x\in\mathbb{R}^n$ is orthogonal to each of these vectors, then
$$x . y=o$$
I couldn't continue, can you help?
Let $y=\beta_1a_1+\cdots+\beta_ka_k$. Then $$\langle y,x\rangle=\langle \beta_1a_1+\cdots+\beta_ka_k,x\rangle=\beta_1\langle a_1,x\rangle+\cdots+\beta_k\langle a_k,x\rangle=\beta_10+\cdots+\beta_k0=0$$ Where I used homogenity and additivity in first slot for the middle equality.