Math 260, Fall 2018, Assignment 11

I have found a very great number of exceedingly beautiful theorems.

- Pierre de Fermat

Carefully define the following terms, then give one example and one non-example of each:

1. $[f]_{\mathcal{A},\mathcal{B}}$ (the matrix of a linear transformation $f:S\rightarrow T$, with respect to a basis $\mathcal{A}$ of $S$ and a basis $\mathcal{B}$ of $T$).

Carefully state the following theorems (you do not need to prove them):

1. Transformation law for vectors (relating $[\vec{v}]_{\mathcal{A}}, S_{\mathcal{A}\rightarrow\mathcal{B}},$ and $[\vec{v}]_{\mathcal{B}}$).
2. Transformation law for matrices (relating $[f]_{\mathcal{A},\mathcal{B}}, S_{\mathcal{A}\rightarrow\mathcal{C}}, S_{\mathcal{B}\rightarrow\mathcal{D}},$ and $[f]_{\mathcal{C},\mathcal{D}}$.)

Solve the following problems:

1. Section 3.4, problems 25 and 27 (in our notation, you are being asked for the matrix $[T]_{\mathcal{B},\mathcal{B}}$; note that the given matrix $A$ is the matrix $[T]_{\mathcal{E},\mathcal{E}}$ of $T$ with respect to the "standard basis" $\mathcal{E}=(\vec{e}_1,\dots,\vec{e}_n)$.)

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