Change of Basis

We have seen in that a linear transformation can be represented by a matrix, once we pick bases for the domain and codomain. How does the matrix representation change if we choose different bases? Which bases lead to especially nice representations? From the infinite possibilities, what is the best possible representation? This section will begin to answer these questions. But first we need to define eigenvalues for linear transformations and the change-of-basis matrix.

1. The change-of-basis matrix is a matrix representation of which linear transformation?
2. B&=\set{\colvector{2\\3},\,\colvector{-1\\2}}& C&=\set{\colvector{1\\0},\,\colvector{1\\1}}
3. surprise,

We now define the notion of an eigenvalue and eigenvector of a linear transformation. It should not be too surprising, especially if you remind yourself of the close relationship between matrices and linear transformations.

We will see shortly the best method for computing the eigenvalues and eigenvectors of a linear transformation, but for now, here are some examples to verify that such things really do exist.

Here is another.

Of course, these examples are meant only to illustrate the definition of eigenvectors and eigenvalues for linear transformations, and therefore beg the question, How would I find eigenvectors? We will have an answer before we finish this section. We need one more construction first.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Given a vector space, we know we can usually find many different bases for the vector space, some nice, some nasty. If we choose a single vector from this vector space, we can build many different representations of the vector by constructing the representations relative to different bases. How are these different representations related to each other? A change-of-basis matrix answers this question.

Notice that this definition is primarily about a single vector space ($V$) and two bases of $V$ ($B$, $C$). The linear transformation ($I_V$) is necessary but not critical. As you might expect, this matrix has something to do with changing bases. Here is the theorem that gives the matrix its name (not the other way around).

So the change-of-basis matrix can be used with matrix multiplication to convert a vector representation of a vector ($\vect{v}$) relative to one basis ($\vectrep{B}{\vect{v}}$) to a representation of the same vector relative to a second basis ($\vectrep{C}{\vect{v}}$).

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Here is the main theorem of this section. It looks a bit involved at first glance, but the proof should make you realize it is not all that complicated. In any event, we are more interested in a special case.

We will be most interested in a special case of this theorem (), but here is an example that illustrates the full generality of .

Here is a special case of the previous theorem, where we choose $U$ and $V$ to be the same vector space, so the matrix representations and the change-of-basis matrices are all square of the same size.

This is the third surprise of this chapter. considers the special case where a linear transformation has the same vector space for the domain and codomain ($V$). We build a matrix representation of $T$ using the basis $B$ simultaneously for both the domain and codomain ($\matrixrep{T}{B}{B}$), and then we build a second matrix representation of $T$, now using the basis $C$ for both the domain and codomain ($\matrixrep{T}{C}{C}$). Then these two representations are related via a similarity transformation () using a change-of-basis matrix ($\cbm{B}{C}$)!

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

tells us that the eigenvalues of a linear transformation are the eigenvalues of any representation, no matter what the choice of the basis $B$ might be. So we could now unambiguously define items such as the characteristic polynomial of a linear transformation, which we would define as the characteristic polynomial of any matrix representation. We will say that again eigenvalues, eigenvectors, and characteristic polynomials are intrinsic properties of a linear transformation, independent of the choice of a basis used to construct a matrix representation.

As a practical matter, how does one compute the eigenvalues and eigenvectors of a linear transformation of the form $\ltdefn{T}{V}{V}$? Choose a nice basis $B$ for $V$, one where the vector representations of the values of the linear transformations necessary for the matrix representation are easy to compute. Construct the matrix representation relative to this basis, and find the eigenvalues and eigenvectors of this matrix using the techniques of . The resulting eigenvalues of the matrix are precisely the eigenvalues of the linear transformation. The eigenvectors of the matrix are column vectors that need to be converted to vectors in $V$ through application of $\ltinverse{\vectrepname{B}}$ (this is part of the content of ).

Now consider the case where the matrix representation of a linear transformation is diagonalizable. The $n$ linearly independent eigenvectors that must exist for the matrix () can be converted (via $\ltinverse{\vectrepname{B}}$) into eigenvectors of the linear transformation. A matrix representation of the linear transformation relative to a basis of eigenvectors will be a diagonal matrix an especially nice representation! Though we did not know it at the time, the diagonalizations of were really about finding especially pleasing matrix representations of linear transformations.

Here are some examples.

Another example, this time a bit larger and with complex eigenvalues.

We finally have enough theorems to understand how Sage creates and manages linear transformations. With a choice of bases for the domain and codomain, a linear transformation can be represented by a matrix. Every interesting property of the linear transformation can be computed from the matrix representation, and we can convert between representations (of vectors and linear transformations) with change-of-basis matrices, similarity and matrix multiplication.

So we can understand the theory of linear algebra better by experimenting with the assistance of Sage, and the theory of linear algebra helps us understand how Sage is designed and functions. A virtuous cycle, if there ever was one. Keep it going.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.