Abstract Linear Algebra 7 | Change of Basis [dark version]

TL;DR

Change of basis converts coordinate vectors when switching from one basis B to another basis C in the same finite-dimensional vector space V.

Briefing Cornell Notes

Briefing

Change of basis is the mechanism for translating the same abstract vector’s coordinates when the underlying basis in a finite-dimensional vector space changes. The practical payoff is immediate: different problems often call for different bases—ones that create lots of zeros in computations or ones that respect a problem’s symmetry—so the key question becomes how to convert coordinates from an “old” basis B to a “new” basis C without redoing everything in the abstract space.

The setup starts with a finite-dimensional vector space V of dimension n. Choosing a basis B = {B1, …, Bn} lets one use the basis isomorphism to identify V with the concrete space F^n (where F is R or C). Under this identification, each basis vector Bj corresponds to a canonical unit vector Ej in F^n. A second basis C = {C1, …, Cn} produces a different basis isomorphism, again mapping V to F^n, but now Cj corresponds to Ej instead. Since both bases represent the same abstract vectors in V, the same vector v has two coordinate descriptions: one as a column vector of coefficients (β1, …, βn) relative to B, and another as (γ1, …, γn) relative to C.

A concrete example uses the polynomial space P2(R), the set of real polynomials of degree at most 2. The standard “monomial” basis is m0(x)=1, m1(x)=x, and m2(x)=x^2. For the polynomial p(x)=3x^2+8x−2, the coordinates relative to {m0,m1,m2} are found by matching coefficients: p = (−2)m0 + 8m1 + 3m2, giving the coordinate vector (−2, 8, 3). Applying the inverse basis isomorphism recovers the abstract polynomial from that coordinate vector.

Next, a different basis C is introduced with three elements: C1=m0, C2=m1, and C3(x)=3x^2+8x. Even though C3 is itself a polynomial combination of m0,m1,m2, it still forms a valid basis once linear independence and spanning are checked. With respect to this new basis, the same polynomial p can be represented more simply: since C3 equals 3x^2+8x, the polynomial p differs from C3 only by the constant term −2, so p = 1·C3 + (−2)·C1 (and the coefficient of C2 is 0). That yields a new coordinate vector (−2, 0, 1) relative to C. The same abstract object now corresponds to different coordinate columns, and the conversion between these columns is exactly what “change of bases” means.

In general terms, the change of basis is constructed as a linear map f: F^n → F^n that converts coordinates in the B-representation directly into coordinates in the C-representation. Conceptually, it is built by composing the inverse of the B-identification with the C-identification: take an old coordinate vector, map it back to V using the inverse of the B basis isomorphism, then map the result into F^n using the C basis isomorphism. Because this is a linear map on F^n, it can be represented by a matrix—an idea reserved for the next step.

Cornell Notes

For a finite-dimensional vector space V, choosing a basis turns abstract vectors into coordinate vectors in F^n. If basis B and basis C are both used to represent the same vector v, then v has two different coordinate columns: β relative to B and γ relative to C. Change of basis is the linear map f: F^n → F^n that converts β into γ directly, without repeatedly working in the abstract space V. It is defined by composing the inverse of the B basis isomorphism with the C basis isomorphism. Since f is linear on F^n, it can be expressed as a matrix, which enables systematic coordinate conversion.

Why does changing the basis matter even when the underlying vector space V stays the same?

Because the same abstract vector v can be written as different linear combinations of basis vectors. Once a basis is fixed, the coefficients in that linear combination become unique coordinates in F^n. Switching from basis B to basis C changes which linear combinations correspond to the coordinate entries, so the coordinate column changes even though v itself does not.

How does the polynomial example show two different coordinate vectors for the same polynomial?

In P2(R), the monomial basis {m0,m1,m2} with m0=1, m1=x, m2=x^2 gives p(x)=3x^2+8x−2 = (−2)m0 + 8m1 + 3m2, so the coordinates are (−2, 8, 3). With the new basis C where C1=m0, C2=m1, and C3=3x^2+8x, the same polynomial becomes p = 1·C3 + (−2)·C1 + 0·C2, so the coordinates are (−2, 0, 1). Same p, different coordinate columns.

What is the conceptual definition of the change-of-basis map f: F^n → F^n?

It converts coordinates from the B-representation to the C-representation by composition: start with an old coordinate vector in F^n, apply the inverse of the B basis isomorphism to interpret it as an element of V, then apply the C basis isomorphism to express that element as coordinates in F^n. This composition yields a linear, bijective map f on F^n.

Why is it valid to treat change of basis as a linear map on F^n?

Both basis isomorphisms are linear maps between V and F^n, and the inverse of a linear isomorphism is also linear. The change-of-basis map is a composition of linear maps, so it remains linear. Therefore it can be represented by a matrix acting on coordinate vectors in F^n.

How does the “direct conversion” idea avoid unnecessary abstract computations?

Instead of repeatedly translating between V and F^n using inverses and forward maps, the change-of-basis map f is designed to take an old coordinate vector β in F^n and output the new coordinate vector γ in F^n in one step. That keeps calculations in the concrete coordinate space.

Review Questions

Given two bases B and C of an n-dimensional vector space V, what does it mean that the coordinate vectors relative to B and C are unique once the bases are fixed?
For p(x)=3x^2+8x−2 in P2(R), compute its coordinate vectors relative to the monomial basis {1, x, x^2} and relative to the basis {m0, m1, 3x^2+8x}.
Describe the change-of-basis map f in terms of composing which two maps, and explain why f must be linear.

Key Points

1
Change of basis converts coordinate vectors when switching from one basis B to another basis C in the same finite-dimensional vector space V.
2
A basis isomorphism identifies V with F^n by mapping each basis vector Bj or Cj to the corresponding canonical unit vector Ej.
3
The same abstract vector v has different coordinate columns (β vs. γ) depending on the chosen basis, even though v itself is unchanged.
4
In P2(R), p(x)=3x^2+8x−2 has coordinates (−2, 8, 3) in the monomial basis {1, x, x^2}.
5
Using the basis {m0, m1, 3x^2+8x}, the same polynomial becomes (−2, 0, 1), illustrating how coordinates can simplify.
6
The change-of-basis map f: F^n → F^n is defined as the composition of the inverse B-identification followed by the C-identification.
7
Because f is linear and bijective on F^n, it can be represented by a matrix for systematic coordinate conversion.

Highlights

Switching bases changes coordinates, not the underlying vector: the same v can correspond to different coordinate columns in F^n.

In P2(R), the polynomial 3x^2+8x−2 becomes (−2, 8, 3) in {1, x, x^2} but (−2, 0, 1) in {1, x, 3x^2+8x}.

Change of basis is built as a direct linear map f: F^n → F^n that converts old coordinates to new ones via a composition of basis isomorphisms.

Since the change-of-basis map is linear on F^n, it admits a matrix representation (set up for the next step).

Topics

Change of Basis
Basis Isomorphism
Coordinate Vectors
Polynomial Vector Spaces
Linear Maps