Basis

The word basis is used often in English speech and its colloquial definition is actually a good way to look at the word basis in linear algebra:

Basis, ba·sis \ ˈbā-səs \ plural bases\ ˈbā-ˌsēz \ Noun

Something on which something else is established or based. Example 1: Stories with little basis in reality. Example 2: No legal basis for a new trial.

The reason for this is that you can choose different bases for a vector space. While the vector space itself does not change when you choose a different basis, the way things are described with numbers does.

Let's look at an example in ℝ2. Consider the vector |u⟩, given as follows:

Figure 4.11 – Graph of the vector |u⟩

Clearly, its coordinates are (3,3). What if I told you I could describe the same vector with the coordinates (3,0)? Wait a minute; that should disturb you. We never talk about the basis in most math...