Mathematics.

vector spaces

Basis and Dimension

Linear Algebra35 minDifficulty5 out of 10

You should know: vector space, linear independence

Overview

A basis of a vector space is a set of vectors that is linearly independent and spans the whole space — every vector in the space can be written in exactly one way as a linear combination of basis vectors. All bases of a given vector space have the same number of elements, and that common number is called the dimension of the space. Dimension is the single most important structural invariant of a vector space: it tells you how many independent 'coordinates' are needed to describe every vector, regardless of which particular basis you choose.

Intuition

A basis is a minimal 'set of building blocks' from which every vector in the space can be assembled, with no redundancy and no gaps. Think of ℝ³: the three axis directions (1,0,0), (0,1,0), (0,0,1) form a basis because any point can be reached by moving some amount along each axis, and no axis direction can be built from the other two. Dimension is just a count of how many independent directions of movement the space has — a plane through the origin in ℝ³ is 2-dimensional even though it lives inside 3-dimensional space, because only two independent directions are needed to describe any point on it.

Formal Definition

Definition

A set B = \{b_1,\dots,b_n\} \subset V is a basis of V if it is linearly independent and spans V:

span⁡(B)=VandB is linearly independent\operatorname{span}(B) = V \quad \text{and} \quad B \text{ is linearly independent}
Basis definition
∀v∈V, ∃! (c1,…,cn):v=c1b1+⋯+cnbn\forall v \in V,\ \exists! \, (c_1,\ldots,c_n): v = c_1 b_1 + \cdots + c_n b_n
Unique representation property
dim⁡(V)=∣B∣(the same for every basis of V)\dim(V) = |B| \quad \text{(the same for every basis of } V \text{)}
Dimension

Worked Examples

  1. Check independence via determinant of the matrix with these as columns.

    det⁡(1101)=1⋅1−1⋅0=1≠0\det\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix} = 1\cdot1 - 1\cdot0 = 1 \neq 0

  2. Since the determinant is nonzero, the two vectors are independent, and 2 independent vectors in a 2-dimensional space automatically span it.

    dim⁡(R2)=2=∣{(1,0),(1,1)}∣\dim(\mathbb{R}^2) = 2 = |\{(1,0),(1,1)\}|

Answer: Yes, it is a basis, since the vectors are independent and there are exactly 2 = dim(ℝ²) of them.

Practice Problems

Difficulty 4/10

What is the dimension of the vector space of all polynomials of degree at most 3?

Difficulty 5/10

Is {(1,1,0), (0,1,1), (1,0,-1)} a basis of ℝ³?

Difficulty 5/10

An image compression algorithm represents each 8×8 pixel block using a basis of 64 standard 'building-block' patterns (as in JPEG's DCT basis). Why must there be exactly 64 basis patterns, not more or fewer?

Quiz

A basis of a vector space V must be:
If dim(V) = n, then any set of n linearly independent vectors in V:
Every vector in V has how many distinct representations as a combination of a fixed basis's vectors?

Summary

  • A basis is a linearly independent spanning set; every basis of a given space has the same size, called the dimension.
  • Every vector has a unique representation as a linear combination of basis vectors (its coordinates).
  • In an n-dimensional space, any n independent vectors automatically form a basis.

References

  1. WebsiteWikipedia — Basis (linear algebra)