Abstract Linear Algebra 7 | Change of Basis

TL;DR

Choosing a basis turns an abstract finite-dimensional vector space V into F^n, making coordinate vectors meaningful.

Briefing Cornell Notes

Briefing

Change of basis is the mechanism for translating the same vector in a finite-dimensional vector space between two different coordinate systems. Since choosing a basis turns an abstract vector space V into a concrete space like F^n (with F = R or C), switching from one basis B to another basis C changes the coordinate column vector—even though the underlying vector v in V stays the same. The practical question becomes: how do the “old” coordinates relative to B relate to the “new” coordinates relative to C?

The setup starts with a basis isomorphism. For a chosen basis B = {b1, …, bn}, the basis isomorphism maps each basis vector bj to the j-th canonical unit vector ej in F^n. Its inverse goes the other way, taking coordinate vectors in F^n back to the corresponding abstract vector in V. This means any v ∈ V can be represented by a coordinate vector in F^n, and the mapping between abstract vectors and coordinate vectors is completely determined once the basis is fixed.

To see why coordinates change, the transcript uses the polynomial space P2(R), consisting of polynomials of degree at most 2. With the standard monomial basis {m0, m1, m2} where m0(x)=1, m1(x)=x, and m2(x)=x^2, a polynomial p(x)=3x^2+8x−2 is written as −2·m0 + 8·m1 + 3·m2. In coordinates relative to this basis, p corresponds to the column vector (−2, 8, 3). Under the basis isomorphism, that coordinate vector is interpreted back as the same polynomial in P2(R).

Next, a different basis C is introduced. Two of its elements match the old ones: c1=m0 and c2=m1, while the third is chosen as c3(x)=3x^2+8x. This new set is still a basis for P2(R), but it changes how p decomposes. In fact, p can be expressed using c3 and the remaining basis vector c1: p = 1·c3 + (−2)·c1 (and c2 is no longer needed). As a result, the coordinate vector of p relative to C differs from (−2, 8, 3), even though p itself has not changed.

The general case formalizes this. Suppose v has coordinates β = (β1, …, βn) in F^n relative to basis B and coordinates γ = (γ1, …, γn) relative to basis C. The vector v in V is the same, but the coordinate vectors differ because the basis vectors used to form linear combinations differ. The change of basis is defined as a linear, bijective map f: F^n → F^n that converts coordinates directly from the B-representation to the C-representation. Concretely, f is built as a composition: first map coordinates back to V using the inverse of the B-based isomorphism, then map into coordinates using the C-based isomorphism. Because f is a linear map from F^n to F^n, it can be represented by a matrix—an observation saved for the next step.

Cornell Notes

With a fixed basis B of a finite-dimensional vector space V, the basis isomorphism identifies V with F^n by sending each basis vector bj to the canonical unit vector ej. Coordinates of a vector v are therefore just the coordinate column in F^n relative to B. Switching to another basis C changes those coordinates even though v itself stays the same. The change of basis is the direct linear bijection f: F^n → F^n that converts B-coordinates to C-coordinates by composing the inverse of the B-isomorphism with the C-isomorphism. Since f is linear on F^n, it can be represented by a matrix, which is the natural next object to compute.

Why can the same abstract vector v have different coordinate vectors?

Coordinates depend on the basis used to express v. In P2(R), the polynomial p(x)=3x^2+8x−2 has coordinates (−2, 8, 3) relative to the monomial basis {1, x, x^2} because p = (−2)·m0 + 8·m1 + 3·m2. After switching to the basis C with c1=m0, c2=m1, and c3(x)=3x^2+8x, the same polynomial can be written as p = 1·c3 + (−2)·c1, so its coordinate vector relative to C is different. The vector in P2(R) is unchanged; only the coordinate representation changes.

How does the basis isomorphism relate abstract vectors to coordinate vectors?

For a basis B = {b1,…,bn}, the basis isomorphism maps bj ↦ ej in F^n, where ej is the j-th canonical unit vector. Its inverse maps a coordinate vector in F^n back to the corresponding abstract vector in V. This means a coordinate vector in F^n uniquely determines the abstract vector once the basis is fixed, and vice versa.

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

The change of basis is defined so it converts coordinates directly without repeatedly moving through V. Given B- and C-based isomorphisms, f is the composition: start with an input coordinate vector x in F^n, apply the inverse of the B-isomorphism to interpret it in V, then apply the C-isomorphism to produce C-coordinates in F^n. This composition is linear and bijective, so it has an inverse and preserves the linear structure of coordinate conversion.

In the polynomial example, why does the new basis make the coordinate expression simpler?

The new basis includes c3(x)=3x^2+8x, which matches most of p(x)=3x^2+8x−2. That alignment lets p be written as p = 1·c3 + (−2)·c1. Under the old monomial basis, p required three coefficients (for 1, x, and x^2). Under the new basis, one basis element already bundles the x^2 and x terms, reducing the number of nonzero coefficients.

What relationship must hold between B-coordinates and C-coordinates for the same vector v?

Both coordinate vectors represent the same abstract vector v in V, meaning they correspond to the same linear combination of basis vectors in V. Once B and C are fixed, the coordinates are uniquely determined, so there is a well-defined conversion rule from β (B-coordinates) to γ (C-coordinates). That conversion rule is exactly the change-of-basis map f.

Review Questions

Given two bases B and C of a finite-dimensional vector space V, what property must the change-of-basis map f: F^n → F^n have, and why?
For p(x)=3x^2+8x−2 in P2(R), write p in coordinates relative to the monomial basis {1, x, x^2} and relative to the basis {m0, m1, 3x^2+8x}.
Explain, using the polynomial example, why choosing a basis tailored to a problem can change the sparsity or simplicity of the coordinate vector.

Key Points

1
Choosing a basis turns an abstract finite-dimensional vector space V into F^n, making coordinate vectors meaningful.
2
Changing from basis B to basis C changes coordinate vectors even though the underlying vector v in V remains the same.
3
The basis isomorphism maps each basis vector bj to the corresponding canonical unit vector ej in F^n, and its inverse maps coordinates back to V.
4
In P2(R), the polynomial p(x)=3x^2+8x−2 has coordinates (−2, 8, 3) in the monomial basis {1, x, x^2}.
5
With the new basis C where c1=m0, c2=m1, and c3(x)=3x^2+8x, the same polynomial becomes p = 1·c3 + (−2)·c1, producing different coordinates.
6
The change of basis is a linear bijection f: F^n → F^n defined by composing the inverse of the B-isomorphism with the C-isomorphism.
7
Because f is a linear map from F^n to F^n, it can be represented by a matrix, which becomes the next computational step.

Highlights

Coordinates change under basis switching even when the vector itself does not—only the coordinate system changes.

In P2(R), embedding the expression 3x^2+8x directly into the basis element c3 makes p(x)=3x^2+8x−2 collapse to p = c3 − 2·c1.

The change-of-basis map is built as a composition: convert B-coordinates back to V, then convert into C-coordinates.

The direct coordinate-to-coordinate conversion f: F^n → F^n is linear and bijective, so a matrix representation is guaranteed.

Topics

Change of Basis
Basis Isomorphism
Coordinate Vectors
Polynomial Bases
Linear Maps