Change of Basis

Earlier on, we introduced the standard basis vectors, and we considered how we could conceptualize of any vector being a linear combination of these vectors. While the standard basis vectors often make life easy by being really simple vectors, it can sometimes be helpful to work in a basis other than the standard basis. We will see one particularly important example of why change of basis is useful later on in Chapter 6.

What Change of Basis Does

We’ve already covered that we can write any vector \(\vec v\) as a linear combination of the basis vectors for the dimension of \(\vec v\). For example, if we consider the case \(\vec v \in \R^2\) (i.e., \(\vec v\) is a vector with two components), we know we can write \(\vec v\) as some combination of \(\transp{[1,0]}\) and \(\transp{[0,1]}\).

There is, however, nothing special about \(\transp{[1,0]}\) and \(\transp{[0,1]}\); \(\vec v\) can be expressed by a linear combination of any two linearly independent vectors in \(\R^2\) (as we know that any two linearly independent vectors in \(\R^2\) span \(\R^2\)).

Example

Consider the two linearly independent vectors \(\vec b_1 = \transp{[1, 2]}\) and \(\vec b_2 = \transp{[3, 1]}\). Let \(B\) be the basis with \(\vec b_1\) as its first coordinate and \(\vec b_2\) as its second coordinate. If we have a vector \(\vec v = \transp{[7,4]}\) with respect to the standard basis (i.e., the “normal” basis we’ve been using this whole time), we could represent in our new basis \(B\) as \(\transp{[1,2]}\).

We can verify this is correct as follows:

\[1\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 2\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 7 \\ 4 \end{bmatrix} = \vec v\]

Just like when we used the standard basis vectors, the first coordinate of our vector \(\transp{[1,2]}\) multiplied the first basis vector, and the second coordinate multiplied the second basis vector.

Definition

We’ll use the notation:

\[[\vec v]_B = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\]

to mean “the vector \(\vec v\) is equal to \(\transp{[1,2]}\) with respect to basis \(B\).”

Hopefully, this makes it clear that the change of basis isn’t really changing the vector in question; rather, it’s just giving us a new way to interpret the coordinates of the vector. An analog from another area of math would be converting between Cartesian/rectangular coordinates and polar coordinates. The points being talked about aren’t changing, but it can be much easier to do certain kinds of problems in polar coordinates as compared to rectangular coordinates.

Computing a Change of Basis

First, we’re going to consider changing between some arbitrary basis \(B\) and the standard basis. It’s possible to construct a change between arbitrary bases by first going to the standard basis and then from the standard basis to any other basis. As such, we’re just going to consider changing between the standard basis

The simple case is converting from the other basis to the standard basis. Let’s call the arbitrary basis \(B\), composed of vectors \(\vec b_1\), \(\vec b_2\), …, \(\vec b_n\). Let \(M\) be the matrix with the basis vectors as its columns, \(M = \left(\vec b_1, \vec b_2, \ldots, \vec b_n\right)\).

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