Basis

Definition

A basis for a Vector Space V\mathbb{V} is a set of vectors that is:

  1. Linearly Independent
  2. Spans / A Generating Set for the entire space V\mathbb{V}

In other words, a basis is a minimal set of vectors that can generate the entire space through Linear Combinations.

Standard Bases

In Rn\mathbb{R}^n

The standard basis consists of vectors with a 1 in one position and 0s elsewhere:

For R2\mathbb{R}^2:

e1=(10),e2=(01)e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}

In Polynomial Space

For polynomials of degree ≤ n:

{1,x,x2,…,xn}\{1, x, x^2, \dots, x^n\}

Properties

  1. Every vector in the space has a unique representation as a linear combination of basis vectors
  2. All bases of a finite-dimensional vector space have the same size (called the Dimension)
  3. Any linearly independent set can be extended to a basis
  4. Any spanning set contains a basis

Why Do We Care?

Bases are fundamental because they:

  1. Give us coordinate systems
  2. Help define dimension
  3. Allow efficient representation of vectors
  4. Simplify computations in the space

Think of a basis as a "coordinate system" for your vector space - it gives you a standard way to describe any vector in terms of simpler pieces.

Exercise

Show that if {v1,v2}{v_1, v_2} is a basis for a vector space V\mathbb{V}, then any vector w∈Vw \in \mathbb{V} can be written uniquely as w=c1v1+c2v2w = c_1v_1 + c_2v_2 for some scalars c1,c2c_1, c_2.