Skip to content

Basis of vector space

1. Basis of a Vector Space

A basis of a vector space is a set of vectors that satisfy two key properties:

  1. Linearly Independent: No vector in the set can be written as a linear combination of the others.
  2. Spans the Space: Every vector in the vector space can be expressed as a linear combination of the basis vectors.

Intuition:

  • Think of the basis as the “building blocks” or “coordinates” of the vector space.
  • For example, in \mathbb{R}^3 , the standard basis is:
    \mathbf{e}_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}, \, \mathbf{e}_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}, \, \mathbf{e}_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}.
    Any vector \mathbf{v} \in \mathbb{R}^3 can be written as:
    \mathbf{v} = c_1 \mathbf{e}_1 + c_2 \mathbf{e}_2 + c_3 \mathbf{e}_3,
    where c_1, c_2, c_3 are scalars.

Properties:

  • A vector space can have infinitely many bases, but all bases have the same number of vectors.
  • The number of vectors in any basis of a vector space is called the dimension of the vector space.

2. Dimension of a Vector Space

The dimension of a vector space is the number of vectors in a basis for the space.

Examples:

  • \mathbb{R}^2 : Dimension is 2 (standard basis is {\mathbf{e}_1, \mathbf{e}_2} ).
  • \mathbb{R}^3 : Dimension is 3 (standard basis is {\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3} ).
  • The space of polynomials of degree \leq n has dimension n+1 (basis: {1, x, x^2, \dots, x^n} ).

Infinite-Dimensional Spaces:

Some vector spaces, such as the space of all continuous functions, have infinitely many basis vectors and are considered infinite-dimensional.

Leave a Reply

error: Content is protected !!