Basis

It is said that the vector w is a linear combination of other vectors, u₁,u₂,…,u_n, if there are k₁,k₂,…,k_nЄR, such that:
w = k₁u₁ + k₂u₂ +….+ k_nu_n

Example: w(3,-3) is linear combination of u(1,1) and v(0,3), because (3,-3)=3(1,1)-2(0,3)

A set of vectors is said to be linearly dependent if one of them is linear combination of the others. Otherwise, the set is said to be linearly independent.

In the plane, two vectors are linearly dependent if and only if they are proportional.

Example: u(1,3),v(-3,-9),w(1,0); u and v are linearly dependent; u and w are linearly independent.

A set of vectors is said to be a generative system if all vectors can be expressed as linear combination of these vectors.
Example: {(1,0),(1,1),(0,1)} is generative system of the plane.

A basis is a set of vectors linearly independent and generative system. If the elements are orthogonal and their magnitudes are 1 (unit vectors), the basis is called orthonormal basis.

Example: B_C={i(1,0),j(0,1)} is an orthonormal basis called canonical basis.