Linear Operators, Representative Matrices and Change of Basis

This is a follow up to a previous question, Change of Basis and Row Operations.

I'm making this concrete to make it a bit easier to write up, but I'm actually interested in the more general case.

Given a linear operator $T:R^3 \rightarrow R^2$ and ordered bases' $B=(b_1,b_2,b_3)$ and $C=(c_1, c_2)$. Let's say I have the images of the basis vectors from $B$, but I don't have any explicit formula for $T$.

Now to get the representative matrix I would just take the coordinates of each image and put them in the appropriate columns like so:

$T=[T]^B_C=\begin{bmatrix} [T(b_1)]_C & [T(b_2)]_C & [T(b_3)]_C \end{bmatrix}$

So here's the follow up question. Let's say I want to convert to new bases without calculating a change of basis matrix and applying it.

Assume that the new input basis can be easily arrived at via elementary operations on $B$. For instance $B'=(b_1+2b_3, b3, b2)$ So I should be able to take:

\begin{bmatrix} b_1 & T(b_1) \\ b_2 & T(b_2) \\ b_3 & T(b_3) \end{bmatrix}

perform row operations like so:

\begin{bmatrix} b_1+2b_3 & T(b_1)+2T(b_3) \\ b_3 & T(b_3) \\ b_2 & T(b_2) \end{bmatrix}

Now I can just transpose the right hand side and take the new coordinate vectors and I've got the new representative matrix:

$T=[T]^{B'}_C=\begin{bmatrix} [T(b_1)+2T(b_3)]_C & [T(b_3)]_C & [T(b_2)]_C \end{bmatrix}$

Also, I think I can get the explicit formula for $T$ according to the new base $B'$ by adding the rows of the row oriented matrix that I did the operations on.

So is all this generally correct? When is it a good idea?

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I was just writing up notes for this in connection with a section on Smith Normal Form in my abstract algebra course. I'll summarize the results here.

Row operations on the matrix change the input basis and column operations on the matrix change the output basis.

First, let $\{u_1, u_2, \ldots, u_n\}$ denote the input basis.

1. Row operation: $r_i \to r_i + k r_j$. Effect: Replace $u_j$ with $u_j - k u_i$. (Note that the $i$ and $j$ switch, and there is a sign change.)

2. Row operation: $r_i \leftrightarrow r_j$. Effect: Swap $u_i$ and $u_j$.

3. Row operation: $r_i \to k r_i$, where $k$ is a unit. Effect: Replace $u_i$ with $k^{-1} u_i$.

Next, let $\{v_1, v_2, \ldots, v_m\}$ be the output basis.

1. Column operation: $c_i \to c_i + k c_j$. Effect: Replace $v_i$ with $v_i - k v_j$. (Here there's just a sign change.)

2. Column operation: $c_i \leftrightarrow c_j$. Effect: Swap $v_i$ and $v_j$.

3. Column operation: $c_i \to k c_i$, where $k$ is a unit. Effect: Replace $v_i$ with $k^{-1} v_i$.

Actually, for Smith Form if you know the Smith form and either the final input basis or the final output basis, you can find the other basis easily. All of this assumes you're doing the easy'' case of a Euclidean domain; you need a fourth type of operation over a PID.

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+1 for interesting, although I have no idea what a Smith Form or PID are. By the way, how correct is what I said above? –  Robert S. Barnes Nov 6 '12 at 21:16
PID: Principle Ideal Domain en.wikipedia.org/wiki/Principal_ideal_domain –  Arkamis Nov 6 '12 at 21:17
I don't quite understand, I've got the input basis and output basis backwards in my example? –  Robert S. Barnes Nov 8 '12 at 8:03

$[T]_{B', C}$ (input basis $B'$, output basis $C$) is the matrix whose columns are formed by applying $T$ to the elements of $B'$, writing the results in components relative to $C$.

You defined the basis $B′$ to be $\{b_1 + 2 b_3, b_3, b_2\}$, so the columns of $[T]_{B', C}$ are, respectively, $$(T(b_1 + 2 b_3))_C = (T(b_1) + 2 T(b_3))_C, \quad (T(b_3))_C, \quad (T(b_2))_C.$$

Those are the columns of the matrix you wrote down, so if I'm interpreting your intent correctly, your matrix is right.

However, I'm not sure what this has to do with row operations. If you told me that (e.g.) $B' = \{FOO, BAR, BAZ\}$, then the columns of $[T]_{B', C}$ are, respectively, $$(T(FOO))_C, \quad (T(BAR))_C, \quad (T(BAZ))_C.$$

I just apply $T$ to whatever you tell me the elements of $B'$ are, wherever they came from, write the results in terms of $C$, and they are the columns of the matrix. It looks as though you originally had a specific problem in mind, thought of a possible method, and are asking about the method without saying what the original problem was. But maybe I'm wrong. I hope this helps.

(Oh --- as for Smith form and PIDs, in linear algebra you work with vector spaces over fields. Fields are where your numbers ("scalars") come from. But you can do a lot of linear-algebra-things with modules [of which vector spaces are a particular case] over rings [of which fields, and PIDs, are a particular case]. So you have matrices and change-of-basis, and things applicable in the general case will work in your case as well. Finally, my linear algebra notes are here: http://www.millersville.edu/~bikenaga/linear-algebra/linear-algebra-notes.html . There are a couple of sections on change-of-base, and change-of-base for linear transformations.)

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