Bases

Level 1 - Math I (Physics) topic page in Vector Spaces.

Principle

A basis is a list of vectors that spans the space and is linearly independent. Spanning means the list can build every vector in the space. Independence means none of the listed vectors is redundant.

For physics, a basis acts like a coordinate system. It lets a vector be described by scalar components without using extra directions that repeat information already present.

Notation

\(B\)

a basis or candidate basis

\(\mathbf b_1,\ldots,\mathbf b_n\)

vectors in the basis

\(\mathbf v\)

a vector being represented

\(c_1,\ldots,c_n\)

scalar coefficients in a basis expansion

\(\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\)

a coordinate representation of \mathbf v using basis vectors

The order of basis vectors matters when coordinates are written as an ordered list.

Method

Check that the candidate list spans the target space.
Check that the candidate list is linearly independent.
If both tests pass, the list is a basis.
To represent a vector, solve \(\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\) for the coefficients.
Use independence to know that the coefficient list is unique.

Rules

Basis representation

\[\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\]

Standard coordinates

\[(x,y)=x(1,0)+y(0,1)\]

Dependent pair

\[(2,0)=2(1,0)\]

A basis must both span the space and be linearly independent.
A basis has no redundant vectors.
Every vector in the space has a unique coefficient list in a chosen basis.
Every basis of a finite-dimensional space has the same number of vectors.

Examples

Question

Why is

\[B=\{(1,0),(0,1)\}\]

a basis for

\[\mathbb R^2\]

Answer

The list spans

\[\mathbb R^2\]

because

\[(x,y)=x(1,0)+y(0,1)\]

for every

\[(x,y)\]

It is independent because

\[c_1(1,0)+c_2(0,1)=(0,0)\]

gives

\[c_1=0\]

and

\[c_2=0\]

Therefore it is a basis.

Checks

Verify both spanning and linear independence.
Do not call every spanning set a basis; redundant spanning sets are not bases.
Basis vectors need not be perpendicular or unit length.
Keep the basis order fixed when writing coordinates.

Spanning Sets

Dimension