Regarding linear dependence and independence for finite sequences of vectors The paragraph in the text—Linear Algebra by Kenneth Hoffman, Ray Kunze—keeps bothering me in that I can't figure out what they are trying to say and that it seems they are simply restating what I've already learnt and that I cannot foresee what is the point in making sure whether $\alpha_1,\alpha_2$ are linearly independent and that the authors leave a mysterious comment—which is the one that I highlighted below—that makes me feel like I'm missing the critical point here. To tell you about what precedes the paragraph the authors had linear dependence and independence defined for finite sequences of vectors and discussed the differences between those of a set of vectors and those of a finite sequence of vectors. The paragraph that I'm struggling with is:

Thus, if $a_1,\cdots,a_n$ are linearly independent, they are distinct and we may talk about the set $\{a_1,\cdots,a_n\}$ and know that it has $n$ vectors in it. So, clearly, no confusion will arise in discussing bases and dimension. The dimension of a finite-dimensional space $V$ is the largest n such that some $n$-tuple of vectors in $V$ is linearly independent—and so on. The reader who feels that this paragraph is much ado about nothing might ask himself whether the vectors
  $$\alpha_1=(e^{\pi/2},1)$$
  $$\alpha_2=(\sqrt[3]{110}, 1)$$
  are linearly independent in $R^2$.

 A: I guess, the thing is, that $e^{\pi/2}=4.8104...\approx\sqrt[3]{110}=4.7914...$, so these vectors are very close to each other, but still independent. Hence they can make a basis in $\mathbb{R}^2$. But yet, I don't grasp the necessity to write that.
The only things, you need to know, is what linearly independent vector set is, what it has to do with the dimension of the space $V$ and the basis in it. So I do not think, you miss something.
A: I think the point is as follows:
Note  $e^{\pi/2}$ and $\sqrt[3]{110}$ are numerically similar, so you $\textbf{might}$ be required to do the work to distinguish them to determine linear dependence.
If
$$\alpha_1 = (e^{\pi/2}, 1)$$
$$\alpha_2 = (\sqrt[3]{110}, 1)$$
and we consider $S = \{\alpha_1, \alpha_2\}$, then under the "set definition" of linear dependence (See definition in Sec. 2.3 p.40), to show linear dependence we must find $\textbf{distinct}$ vectors $\beta_1,..., \beta_n$, in $S$ and scalars $c_1,...,c_n$ in $\mathbb{F}$ not all $0$, such that 
$$ c_1 \beta_1 + \cdots + c_n \beta_n = 0$$ 
So if $\alpha_1 = \alpha_2$, $S$ is still linearly independent.
But clearly this is not the case under the "tuple definition".
In summary, under the set definition, you don't have to do the work of determining whether $\alpha_1 = \alpha_2$ since in either case $S$ will be independent.
But for the tuple $T = (\alpha_1, \alpha_2)$, you do have to do the work to distinguish $\alpha_1$ and $\alpha_2$.
