Vector notation for $kn$ vectors in $Bbb{R^n}$

Question

I'm reading Advanced calculus of several variables by Edwards, C.Henry. On page 6, the author is proceeding with the proof that any $n+1$ vectors in $Bbb{R^n}$ are linearly dependent and writes :
"Suppose that $v_1,...,v_k$ are $k>n$ vectors in $Bbb{R^n}$ and write $$v_j = (a_{1j},...,a_{nj})$$
where $j=1,...,k$."
I have understood vectors of $Bbb{R^n}$, as $n$-tuples of real numbers as in $(x_1, ... , x_n)$. What is the purpose of introducing row column notation to describe them? I think it might have something to do with the next part ;
"...We want to find real numbers $x_1,...,x_k$, not all zero, such that
$bar0=x_1v_1 + x_2v_2 +  ... + x_kv_k$."
Afterwards this vector equation can be written as a system of linear equations with more unknowns than equations ; a non-trivial solution exists and the proof is complete.

Robert Lee · Answer

I don't think the notation means row column vector. It's just a way to uniquely identify each real number. In other words, $a_{ij} in mathbb{R}$ is just the $i$-th entry of the $j$-th vector. This is useful because it easily lets you see the total number of unknowns.
If we have $k$ vectors in $mathbb{R}^n$, then this notation implies $i in mathbb{N}cap [1, n]$ and $j in mathbb{N}cap [1, k]$. Since in the proof we want to see how many equations vs unknowns we have, using this notation we can expand the equation $vec{0} = sum_{i=1}^{k} x_i v_i$, and obtain $n$ equations of the form
$$
0 = x_1 a_{i 1} +  x_2 a_{i 2} + ... + x_k a_{i k} quad  1le i le n
$$
but since $k>n$, here we see that indeed we have more unknowns (the $x_k$'s) than equations.

Vector notation for $k>n$ vectors in $Bbb{R^n}$

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