Bases

A basis to a space is a chosen set of vectors so that any vector in the space can be uniquely expressed in terms of the basis vectors:

Example: are a basis to coordinate space:

Basis vectors must be independent, which means you need all of them to express any arbitrary vector in the space. If some basis vector can be expressed in terms of the others, you do not need that vector and should throw it out.

For example if would be , then you would not need it, since could then be written as . But there is no way to get from a linear combination of and

To check independence of supposed basis vector , verify that

only when all coefficients are zero.

Why this works: If, for example, c₁ would be nonzero, you can take to the other side and divide by -c₁.