A Basis for a Vector Space

Let V be a subspace of R ⁿ for some n. A collection B = { v₁, v₂, …, v _r } of vectors from V is said to be a basis for V if B is linearly independent and spans V. If either one of these criterial is not satisfied, then the collection is not a basis for V. If a collection of vectors spans V, then it contains enough vectors so that every vector in V can be written as a linear combination of those in the collection. If the collection is linearly independent, then it doesn't contain so many vectors that some become dependent on the others. Intuitively, then, a basis has just the right size: It's big enough to span the space but not so big as to be dependent.

Example 1: The collection { i, j} is a basis for R², since it spans R² and the vectors i and j are linearly independent (because neither is a multiple of the other). This is called the standard basis for R². Similarly, the set { i, j, k} is called the standard basis for R³, and, in general,
is the standard basis for R ⁿ.

I hope the above explanation was useful.