Change of basis基底変換きていへんかん and similarity相似そうじ

1Introduction

Changing a basis基底きてい does not change a vectorベクトル itself, but it changes its coordinate representation座標表示ざひょうひょうじ. The same is true for a linear map線型写像せんけいしゃぞう: the map itself is unchanged, while its representation matrix表現行列ひょうげんぎょうれつ changes.

This distinction is essential in linear algebra線型代数せんけいだいすう. When looking at a matrix, one must distinguish the object itself from the representation obtained after choosing a basis. Without this distinction, the meaning of diagonalization対角化たいかくか and eigenvalues固有値こゆうち becomes obscure.

Diagonalization and eigenvalues are introduced later. In this lecture they are mentioned only as a preview of why representation matters.

2Terms and definitions

Change of basis基底変換きていへんかん means rewriting the same vector or the same linear map using coordinates with respect to a different basis.

Let $P$P be the invertible matrix satisfying

In this lecture, $P$P converts coordinates in the new basis into coordinates in the old basis.

Two square matrices $A$A and $B$B are similar相似そうじ if there is an invertible matrix $P$P such that

Similar matrices represent the same linear map in different bases.

3Plan

For change of basis, first fix what changes and what stays invariant.

The eigenvalue and characteristic-polynomial entries in this table are previews. The main content of this page is coordinate representation, representation matrices, and invertible coordinate changes.

4Intuitive explanation

The same point in the plane has different coordinates depending on whether it is measured with horizontal and vertical axes or with slanted axes. The point has not moved; only the coordinate system has changed.

A linear map behaves the same way. The same rotation or stretch may have a different matrix form after changing the basis. Diagonalization対角化たいかくか uses this idea aggressively: if eigenvectors can be chosen as a basis, the map is represented by a diagonal matrix of scale factors.

5Precise explanation

5.11. Changing coordinates of a vector

Choose an old basis and a new basis for the same space $V$V. Define $P$P by

This formula translates coefficients in the new basis into coefficients in the old basis. The matrix $P$P must be invertible because coordinates must be recoverable uniquely; a change from one basis to another cannot lose information.

5.22. Changing a representation matrix

Let $T:V\to V$T:V\to V be a linear map線型写像せんけいしゃぞう, and let $A_{\mathrm{o}\mathrm{l}\mathrm{d}}$A_{\mathrm{old}} be its representation matrix in the old basis:

Starting from new coordinates, we have

After applying $T$T, we convert back to new coordinates by multiplying by $P^{-1}$P^{-1}:

Therefore

5.33. Changing domain and codomain bases separately

For a map $T:V\to W$T:V\to W, the domain basis and codomain basis may be changed independently. If

on the input side and

on the output side, then the new representation matrix is

The right factor $P$P translates the input into old coordinates, while the left factor $Q^{-1}$Q^{-1} translates the output back into new coordinates.

6Example: representation in a nonstandard basis

Consider the linear map on $R^2$\mathbb R^2

In the standard basis,

Let

Then

and

Thus

The map has not changed; the basis and representation matrix have changed.

7What similarity preserves

If $B=P^{-1}AP$B=P^{-1}AP, then $A$A and $B$B are two coordinate representations of the same linear map. Hence properties independent of the choice of basis are preserved.

In the following table, the rows about eigenvalues, characteristic polynomials, and determinants are previews for later lectures. Here they should be read as examples of the principle that coordinate-independent quantities are preserved under a change of representation.

Matrix entries, columns themselves, and the visual picture in the standard basis may change. Similarity means viewing the same map in different coordinates, not keeping components fixed.

8Scope and limitations

Similarity is for square matrices representing maps from a space to itself. For rectangular matrices, the domain and codomain bases change separately, producing $Q^{-1}AP$Q^{-1}AP rather than $P^{-1}AP$P^{-1}AP. Whenever $P^{-1}AP$P^{-1}AP is used, $P$P must be invertible; otherwise it is not a basis-to-basis coordinate transformation.

If $P$P is not invertible, it collapses information instead of changing from one basis to another, so the expression no longer represents a genuine change of coordinates.

9Final forms

10In one sentence

Change of basis基底変換きていへんかん changes coordinate representations, while similar相似そうじ matrices represent the same linear map線型写像せんけいしゃぞう in different bases.

• A change of basis基底変換きていへんかん changes coordinate representations, not the vector or linear map itself.
• With the convention $[x{]}_{\mathrm{o}\mathrm{l}\mathrm{d}}=P[x{]}_{\mathrm{n}\mathrm{e}\mathrm{w}}$[x]_{\mathrm{old}}=P[x]_{\mathrm{new}}, the new representation of $T:V\to V$T:V\to V is $A_{\mathrm{n}\mathrm{e}\mathrm{w}}=P^{-1}{A}_{\mathrm{o}\mathrm{l}\mathrm{d}}P$A_{\mathrm{new}}=P^{-1}A_{\mathrm{old}}P.
• Similar相似そうじ matrices represent the same linear map in different bases, so basis-independent quantities such as eigenvalues, characteristic polynomial, determinant, rank, and invertibility are preserved.
• For a map $T:V\to W$T:V\to W, domain and codomain bases are changed separately, so the correct form is $Q^{-1}{A}_{\mathrm{o}\mathrm{l}\mathrm{d}}P$Q^{-1}A_{\mathrm{old}}P, not ordinary similarity.

Eigenvalues, characteristic polynomials, and determinants are defined and proved later. In this summary they are previews illustrating that similarity is a different representation of the same object.

11Exercise links

12Related links