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In a few situations, I found myself being asked by younger students why the concepts of a basis was important.

First, in the concept of linear spaces, it's easy to explain that having a basis allows us to describe the spaces with a relatively small set and makes working with linear transformation a much easier task (there are many more reasons, but this was an introductory linear algebra course).

When I was asked the same question in the context of propositional logic (a basis here is a maximally independent set of formulas), I found it a bit more difficult to motivate the definition.

In general, when asked "What is the importance of a basis?", what do you reply?

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  • $\begingroup$ The most basic question in linear algebra is "describe the general solution to $Ax=b$." Doing that amounts to writing $x=x_0+\sum_{i=1}^n c_i k_i$, where $c_i$ are arbitrary scalars and $k_i$ make a basis for the kernel of $A$. Each choice of the $c_i$ gives a different solution. So the meaningful information here is the particular solution $x_0$ and the basis $k_i$ for the kernel. They already know this fact before they get to vector spaces in the first place; it is just formal language for something they already understand. $\endgroup$ – Ian Apr 9 '16 at 21:18
  • $\begingroup$ I think putting everything in this framework will help them, especially if they cannot easily understand why a statement like "a linear transformation is determined by its action on a basis" is useful. $\endgroup$ – Ian Apr 9 '16 at 21:20
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In every context, a basis is essentially a (not the) smallest set you need to understand, in order to understand the whole object in question.

For example, a linear transformation is determined by its action on a basis. Similarly, a complete theory is determined by which elements of a (propositional) basis it contains.

The virtue of the word "smallest" in the first paragraph is that, roughly speaking, every behavior over a basis is possible. For example, if $\{b_i\}$ ($i\in I$) is a basis for $V$ and $w_i$ $(i\in I)$ are vectors in $W$, then there is a linear transformation $T$ sending $b_i$ to $w_i$ (and by the above paragraph, it's unique). Similarly, given a propositional basis $p_i$ $(i\in I$) and a map $t: I\rightarrow \{\top, \bot\}$, there is a complete theory $T$ such that $p_i\in T\iff t(i)=\top$ (and by the above, it's unique).

These two aspects of bases (determination and freedom) mean that they're invaluable for understanding the things that act (in a very loose sense) on them - linear transformations, or complete theories, or etc.

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