In three-space, we use the set of vectors {i, j, k} to describe all the other vectors in the space. This set of vectors is ideal for the job for two reasons: you can get all the vectors in 3-space by taking linear combinations of i, j and k (i.e. they span all of the space) and you need all three; you can't replace any of them by a linear combination of the others (i.e. they are linearly independent). Similarly, in 2-space, the set of vectors {i, j} is ideal for describing all vectors.

In this learning object, we're going to look for other optimal sets of vectors for 2-space and 3-space called bases. Specifically, we'll

1. Look at how to define a basis,
2. Define coordinate systems related to bases,
3. Look at which bases allow us to use the usual coordinate rules for dot products and related quantities.