linear algebra – Show the maximum number of linearly independent $n$-vectors is $n$

Show the maximum number of $n$-vectors that can be linearly independent is $n$.

$n$ referring to the order of the vector, take $n$ linearly independent $n$-vectors. Take any other vector. It’s possible to write the vector as a linear combination of the $n$ linearly independent vectors, so $v_{n+1}=a_1v_1+a_2v_2+dots+a_nv_n$. Then we have $a_1v_1+a_2v_2+dots+a_nv_n-v_{n+1}=0$ and the vectors are linearly dependent. Thus $n$ is the maximum number of linearly independent $n$-vectors.

I apologize if this is trivial, but I wanted to verify it nonetheless.