Basis Vectors

Introduction to basis vectors:

Definition

Consider the non-zero vectors v1, v2 …, vn in the vector space V over the field F. We call them basis vectors if they satisfy the following two conditions:

The vectors v1, v2 …, vn are linearly independent
Any vector v in the vector space V can be written as a linear combination of the basis vectors. In other words, there exists scalars a1, a2…, an not all zero such that v = a1v1 + a2v2 +…+ anvn I like to share this What are Vectors with you all through my article.


On satisfying the above two conditions we call the set B = {v1, v2 …, vn } as the basis of the vector space V and the elements of B are called the basis vectors.

Example for Basis Vectors

Let V = R3 and consider the vectors:

E1 = (1, 0, 0)

E2 = (0, 1, 0)

E3 = (0, 0, 1)

We will show that the above vectors form the basis of R3.

Let a, b, c be real numbers so that aE1 + bE2 + cE3 = 0

Then a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) = (0, 0, 0)

Implies (a, b, c) = (0, 0, 0) which means a = b = c = 0

Hence they are linearly independent vectors.

Also any vector v = (x, y, z) in V can be written as a linear combination of E1, E2, E3 as follows: v = xE1 + yE2 + zE3

Hence {E1, E2, E3} are the basis vectors for R3.

Note: We call the above set of basis vectors as the standard basis of R3. It can be extended to Rn where the standard basis vectors are E1, E2…, En.

Remember:  A vector space V is said to be of finite dimension or finitely generated if there exists a finite number of basis vectors in V else it is referred to as infinite dimensional space. Please express your views of this topic math problems 8th grade by commenting on blog.

Exercise to Basis Vectors:

Show that the infinite set S = {1, x, x2, …, xn,...} forms the basis vectors of the vector space F[x] of polynomials over the field F.