Can two $n$-dimensional vectors that are not scalar multiples of each other be added to make any point on an $n$-dimensional coordinate plan?

Question

If we have $\pmb{u}, \pmb{v} \in \mathbb{R} ^ n$ and $c\pmb{u} \ne \pmb{v}$ where $c \in \mathbb{R}$, then is

$\forall_{a_1, \ldots, a_n \in \mathbb{R}} \left (\exists_{m, n \in \mathbb{R}} \left (m\pmb{u} + n\pmb{v} = \begin{bmatrix} a_1 \\ \vdots \\ a_n \\ \end{bmatrix} \right ) \right ) \text{ true?}$

Note: Bolded letters represent vectors.

Answer 1

We know that $\mathbb{R^n}$ has a basis of $n$ vectors (such as the standard unit basis {$e_1, e_2,.... e_n$}). By a known theorem from linear algebra, we then have that any set of less than $n$ vectors in $\mathbb{R^n}$ does not span $\mathbb{R^n}$. Hence, as long as $n > 2$, vectors $u$ and $v$ could not span $\mathbb{R^n}$. That is, $\exists$ $a_1...a_n$ such that $\forall m,n \in \mathbb{R}$, $mu + nv \neq (a_1...a_n)$, which means the given statement cannot be true.

Answer 2

The linear span of two vectors can never be more than a plane.