# Change of basis vs linear transformation

31 May 2016There are two related concepts in linear algebra that may seem confusing
at first glance: **change of basis** and **linear transformation**.
Change of basis formula relates coordinates of one and the same vector
in two different bases, whereas a linear transformation relates coordinates
of two different vectors in the same basis. The difficulty in discerning
these two cases stems from the fact that the word *vector* is often
misleadingly used to mean *coordinates of a vector*. Generally speaking,
$\newcommand{\vec}{\mathbf} \vec{x} \neq (x_1, x_2)^T$,
unless a certain basis is understood.

## Change of basis

Let’s consider a concrete example. Let $(\vec{u}_1, \vec{u}_2)$ be an orthonormal basis in $\mathbb{R}^2$. Imagine we make a copy of it $(\vec{v}_1, \vec{v}_2)$ and rotate the copy by $\theta$ degrees.

Without loss of generality, we can identify the initial basis vectors
with the *standard unit vectors* of $\mathbb{R}^2$:

Now, vectors $\vec{v}_1$ and $\vec{v}_2$ can be easily represented in the basis $(\vec{u}_1, \vec{u}_2)$ as

More compactly, one writes

The rotation matrix on the right-hand side relates bases $(\vec{u}_1, \vec{u}_2)$ and $(\vec{v}_1, \vec{v}_2)$. In general, change of basis in $\mathbb{R}^2$ is described by the formula

where $(\vec{u}_1, \vec{u}_2)$ is an old basis, $(\vec{v}_1, \vec{v}_2)$ is a new basis, and matrix $(\vec{u} \to \vec{v})$ specifies a relationship between them.

## Change of coordinates of a vector

A vector is an object that exists independent of a basis. Although it is common in engineering and mathematics to write $\vec{x} = (x_1, x_2)^T$, one should be aware that this notation implies a certain choice of a basis; namely, it implies that the standard basis of $\mathbb{R}^2$ is chosen. That is,

with $(\vec{u}_1, \vec{u}_2)$ from \eqref{standard_basis}.

Let’s consider coordinates of $\vec{x}$ in basis $(\vec{v}_1, \vec{v}_2)$. Analogously to \eqref{vec_in_u},

Equating expansions of $\vec{x}$ \eqref{vec_in_u} and \eqref{vec_in_v} while substituting \eqref{change_of_basis} in place of $(\vec{v}_1, \vec{v}_2)$, we obtain

On both sides, we have expansions of $\vec{x}$ in
basis $(\vec{u}_1, \vec{u}_2)$, therefore coordinates on both sides
should be equal. Thus, we arrive at the formula for
the *change of coordinates of a vector* under change of basis

whith coordinates of $\vec{x}$ in the old basis $\vec{x}^\vec{u} = (x_1, x_2)^T$, and coordinates of $\vec{x}$ in the new basis $\vec{x}^\vec{v} = (x’_1, x’_2)^T$.

## Linear transformation

In contrast to the previous section, we now fix the basis $(\vec{u}_1, \vec{u}_2)$ and represent all vectors in that basis. The question we want to answer is “How to represent a linear transformation $\newcommand{\bphi}{\boldsymbol{\phi}} \bphi : \mathbb{R}^2 \to \mathbb{R}^2$ by a matrix?”

Let’s apply $\bphi$ to some vector $\vec{x}$:

By expanding $\vec{y}$ in basis $(\vec{u}_1, \vec{u}_2)$ and rewriting the right-hand side as matrix-vector multiplication, we obtain

Now we are approaching the point where confusion arises. Assume $\bphi$ rotates every vector by $\theta$. Then the matrix representation of $\bphi$ is precisely the matrix $(\vec{u} \to \vec{v})$ we had before. Therefore,

where we identify vectors with their coordinates in the standard basis as conventionally done in sciences (i.e., $\vec{y} = \vec{y}^\vec{u}$ and $\vec{x} = \vec{x}^\vec{u}$).

Compare formulas \eqref{change_of_coordinates} and \eqref{linear_transformation}. They look very similar as they both relate two column vectors via the same matrix. There is, however, a big difference between them. Equation \eqref{change_of_coordinates} expresses the coordinates of $\vec{x}$ in the old reference frame given its coordinates in the new one, whereas equation \eqref{linear_transformation} expresses the coordinates of the transformed vector $\vec{x}$ given the coordinates of the untransformed vector $\vec{x}$—all in one reference frame. We could also invert \eqref{change_of_coordinates} to always have new coordinates on the left-hand side, $\vec{x}^\vec{v} = (\vec{u} \to \vec{v})^{-1} \vec{x}^\vec{u}$. In this form, the meaning of the difference between \eqref{change_of_coordinates} and \eqref{linear_transformation} becomes clear.

The best strategy to avoid mistakes is to pick one of the two possibilities—either transform bases or transform vectors—and stick with it.

## How transformations transform

Let’s have a look at how linear transformations transform under a change of basis. Notation in this section slightly differs from the rest of the article; namely, we use primed symbols to denote objects related to a new basis.

Consider a basis transformation $\vec{u} = \vec{u}’ \vec{T}^{-1}$, where $\vec{u}$ is the old basis and $\vec{u}’$ is the new basis. Then,

or, in tensor notation,