30. Basis

Non-redundant set of vectors that span

A basis of a vector space is a set of vectors that satisfies two conditions:

  1. Linear Independence: no vector in the set can be written as a linear combination of the others — equivalently, the only solution to

    is .

  2. Spanning: every vector can be expressed as a linear combination of the basis vectors:

Example

Consider the vector space (the 2-dimensional Euclidean space). A common basis for is , where:

This set is a basis because:

  • Linear Independence: The only solution to is .
  • Spanning: Any vector can be written as

This means is a basis for , and the dimension of is 2.