Kuzee
Vector Algebra and Analytic Geometry of Space

Bases and Dimension

The Definition of a Basis

Basis

A basis of the vector space VV is a linearly independent set of vectors in VV such that every vector in VV can be written as a linear combination of this set.

Properties of a Basis

  1. Linear independence: the vectors of a basis are linearly independent as a whole (not merely pairwise linearly independent)
  2. Spanning: a basis spans the whole vector space
  3. Minimality: removing any one of them fails to span the whole space

The Standard Basis

The standard basis of an n-dimensional space

In Rn\mathbb{R}^n, the standard basis is:

e⃗1=(1,0,0,…,0)\vec{e}_1 = (1, 0, 0, \dots, 0) e⃗2=(0,1,0,…,0)\vec{e}_2 = (0, 1, 0, \dots, 0) ⋮\vdots e⃗n=(0,0,0,…,1)\vec{e}_n = (0, 0, 0, \dots, 1)

The Dimension Theorem

Theorem1

Any two bases of the vector space VV contain the same number of vectors. This number is called the dimension of VV, written dim⁡V\dim V.

What the Dimension Means

  • a finite-dimensional space: a vector space with a finite basis
  • an infinite-dimensional space: a vector space with no finite basis
  • a zero-dimensional space: a space containing only the zero vector, with dim⁡V=0\dim V = 0

The Dimensions of Common Vector Spaces

  • the dimension of Rn\mathbb{R}^n is nn
  • the dimension of the zero vector space is 00
  • the dimension of the polynomial space PnP_n is n+1n+1

The Coordinate Representation

The Concept of Coordinates

Coordinates

Let {e⃗1,e⃗2,…,e⃗n}\{\vec{e}_1, \vec{e}_2, \dots, \vec{e}_n\} be a basis of VV. Then every vector a⃗∈V\vec{a} \in V can be written uniquely as:

a⃗=x1e⃗1+x2e⃗2+⋯+xne⃗n\vec{a} = x_1\vec{e}_1 + x_2\vec{e}_2 + \dots + x_n\vec{e}_n

where x1,x2,…,xnx_1, x_2, \dots, x_n are called the coordinates of a⃗\vec{a} in the basis {e⃗1,e⃗2,…,e⃗n}\{\vec{e}_1, \vec{e}_2, \dots, \vec{e}_n\}.

The Coordinate Vector

The coordinate vector

a⃗=(x1x2⋮xn){e⃗1,e⃗2,…,e⃗n}\vec{a} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}_{\{\vec{e}_1, \vec{e}_2, \dots, \vec{e}_n\}}

Change of Basis

The Concept of a Change of Basis

Change of basis

When we move from one basis to another, the coordinates of a vector change as well.

Suppose VV has two bases: Basis I: {e⃗1,e⃗2,…,e⃗n}\{\vec{e}_1, \vec{e}_2, \dots, \vec{e}_n\} Basis II: {e⃗1′,e⃗2′,…,e⃗n′}\{\vec{e}'_1, \vec{e}'_2, \dots, \vec{e}'_n\}

Then there is a transition matrix PP such that:

(e⃗1′e⃗2′…e⃗n′)=(e⃗1e⃗2…e⃗n)P\begin{pmatrix} \vec{e}'_1 & \vec{e}'_2 & \dots & \vec{e}'_n \end{pmatrix} = \begin{pmatrix} \vec{e}_1 & \vec{e}_2 & \dots & \vec{e}_n \end{pmatrix} P

Exercises

Exercise 1

Prove that the dimension of R2\mathbb{R}^2 is 2.

vector spacedimensionproperties of a basis

Approach:

Prove that a basis containing 2 vectors exists, and that no basis containing 1 vector exists.

Steps:

  1. A 2-dimensional basis exists: {(1,0),(0,1)}\{(1,0), (0,1)\} is a basis of R2\mathbb{R}^2

    • it is linearly independent
    • it spans every vector: (a,b)=a(1,0)+b(0,1)(a,b) = a(1,0) + b(0,1)

  2. No 1-dimensional basis exists: suppose there were a 1-dimensional basis {u⃗}\{ \vec{u} \}

    • u⃗≠0⃗\vec{u} \neq \vec{0}
    • but it cannot span vectors such as (1,0)(1,0) (if u⃗=(1,0)\vec{u} = (1,0), it cannot span (0,1)(0,1))

Answer: dim⁡R2=2\dim \mathbb{R}^2 = 2

Exercise 2

Find the coordinates of the vector a⃗=(1,2,3)\vec{a} = (1, 2, 3) in the basis {(1,0,1),(0,1,1),(1,1,0)}\{(1,0,1), (0,1,1), (1,1,0)\}.

the coordinate representationchange of basisa system of equations

Approach:

Solve a system of equations for the coordinates.

Steps:

Put a⃗=xe⃗1+ye⃗2+ze⃗3\vec{a} = x\vec{e}_1 + y\vec{e}_2 + z\vec{e}_3

That is: (1,2,3)=x(1,0,1)+y(0,1,1)+z(1,1,0)(1,2,3) = x(1,0,1) + y(0,1,1) + z(1,1,0) =(x+z,y+z,x+y)= (x+z, y+z, x+y)

Solve the system: {x+z=1y+z=2x+y=3\begin{cases} x + z = 1 \\ y + z = 2 \\ x + y = 3 \end{cases}

Solving gives: x=1x = 1, y=2y = 2, z=0z = 0

Answer: the coordinates are (1,2,0)(1, 2, 0)

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
dim⁡V\dim VDimensiondimension of VThe dimension of the vector space VV
e⃗1,e⃗2,…\vec{e}_1, \vec{e}_2, \dotsBasis vectorbasis vectorsThe vectors of a basis
PPTransition matrixtransition matrixThe matrix of a change of basis
(x1,x2,…,xn)(x_1, x_2, \dots, x_n)CoordinatescoordinatesThe coordinates of a vector in a basis
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The Definition of a Vector Space
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Coordinate Transformations

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