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What does it mean to find the matrix of a linear transformation with respect to a basis in the domain, and another in the codomain? In our notes we usually use the standard basis in the domain and a non-standard basis in the codomain. I assume the solution differs in we use a non-standard basis in the domain, but I'm not really sure how.

I'm probably not being very clear so I'll include an example.

Consider the basis $\beta = \{1,t,t^2\}$ for $P_2(\mathbb{R})$. Let $D: P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ be defined by differentation, $D(p)(t) = p'(t)$. Find the matrix $B_D$ of $D$ with respect to $\beta$ in the domain and $C = \{1+t, t-t^2, t^2\}$ in the codomain.

I know how to answer this question, but not really what it means. How would it differ if $\beta$ wasn't the standard basis? How do you do it with respect to two different bases?

Some clarification would be much appreciated, thanks!

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up vote 1 down vote accepted

An important point of abstract linear algebra is that "bases are arbitrary"; a "standard basis" is just for convenience and has no mathematical significance. The way you answer this question doesn't change at all no matter what bases (standard or not) that you choose for the domain or codomain:

  1. Find the image, under $D$, of each element of the given basis for the domain.
  2. Write these images in terms of the given basis for the codomain. This gives you a set of coordinates (real numbers, in your example) for each element in the image.
  3. Collect these coordinates into a matrix.
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Thanks, I was confusing myself with what we were actually doing in the first step. All clear now. –  user1520427 Sep 18 '12 at 5:08
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