This isn't a homework problem, but rather a bit of confusion regarding something in the textbook we're using; if this isn't the right place, feel free to move it.(adsbygoogle = window.adsbygoogle || []).push({});

From Artin'sAlgebrapages 422/423 (slightly paraphrased):

Let ##Q=\begin{bmatrix}1&\\3&1\end{bmatrix}##, ##A=\begin{bmatrix}2&-1\\1&2\end{bmatrix}##, ##P=\begin{bmatrix}1&1\\1&2\end{bmatrix}##, ##A'=Q^{-1}AP=\begin{bmatrix}1&\\&5\end{bmatrix}## (note: blank spaces are to be interpreted as zeroes.)

Let ##M## be the integer lattice with its standard basis ##{\bf C}=\left(e_1,e_2\right)##, and let ##L## be the lattice with basis ##{\bf B}=\left(v_1,v_2\right)=\left(\left(2,1\right)^t,\left(-1,2\right)^t\right)##. Its coordinate vectors are the columns of ##A##. We interpret ##P## as the matrix of a change of basis in ##L##, and ##Q## as the matrix of change of basis in ##M##.

My question is, if ##A## is interpreted as a map ##M\rightarrow L##, wouldn't we have to interpret ##P## as a map ##M\rightarrow M## and ##Q## as a map ##L\rightarrow L## to get ##A'##, which is just ##A## with different bases for ##M## and ##L##, to be ##M\rightarrow L##? Therefore ##P## would be interpreted as a change of basis in ##M## instead of ##L## and ##Q## in ##L## instead of ##M##? In fact, if we had set up the problem so ##A## was 2x3, ##Q## 2x2, and ##P## 3x3, ##M## would be ##\mathbb{R}^3## and ##L## would be ##\subseteq\mathbb{R}^2##, so ##P## couldnotbe interpreted as a change of basis in ##L=\mathbb{R}^2## as it's 3x3.

Is there something incredibly obvious I'm missing?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Artin - domains don't seem to match

Loading...

Similar Threads for Artin domains don't |
---|

I Finite Integral Domains ... Adkins & Weintraub, Propn 1.5 |

I How to find admissible functions for a domain? |

**Physics Forums | Science Articles, Homework Help, Discussion**