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I have a question regarding linear algebra and some change of basis. I have 4 matrices A, P, Q and B that represent $f: F^3 -> F^2$, with basis $(u_1,u_2,u_3)$ for the domain and the basis $(v_1,v_2)$ for the co-domain. I have to show the connection between A, P, Q and B.

Consider the linear transformation $f:F^3\rightarrow F^2$ given by: $$ f\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}x_1+2x_2+3x_3\\-x_1+x_2-x_3\end{bmatrix} $$ Where a basis for $F^3$ is $$ u_1=\begin{bmatrix}-1\\1\\-2\end{bmatrix}\quad u_2=\begin{bmatrix}3\\-1\\1\end{bmatrix}\quad u_3=\begin{bmatrix}-1\\0\\1\end{bmatrix} $$ and the basis for $F^2$ is $$ v_1=\begin{bmatrix}4\\3\end{bmatrix}\quad v_2=\begin{bmatrix}5\\4\end{bmatrix}. $$ using the matrices $$ A=\begin{bmatrix}1&2&3\\-1&1&-1\end{bmatrix}\quad P=\begin{bmatrix}-1&3&-1\\1&-1&0\\-2&1&1\end{bmatrix}\quad Q=\begin{bmatrix}4&5\\3&4\end{bmatrix} $$

Editor's note: The $-1$ in the first column and second row of $P$ should likely be $1$, so that the first column is $u_1$, but this formulation matches the data in the OP's link.

This is what I have done so far:

We know that $M_fx = f(x)$, where $M_f$ is expressed in domain K, and the co-domain is expressed in the K'. x is expressed in basis K, and $f(x)$ is expressed in basis K'.

Besides: if x in basis K= x' in basis K', then x=Qx'

we also know that $Bx = f(x)$, where $(u_1, u_2,u_3)$ is the basis for $F^3$ and $(v_1,v_2)$ is the basis for $F^2$. and if $x' = P^{-1}$, then $Bx'=BP^{-1}x = f(x)$ $->$ $BP^{-1}x = f(x) = Q^{-1} * f(x)= Q^{-1}Ax$ $-> BP^{-1}x = Q^{-1}Ax -> B = Q^{-1}AP$

We will try and test if it's true: First we find $Q^{-1}$:

$$ Q^{-1}= 1/(16-15) * {\begin{bmatrix}4&-3\\-5&4\end{bmatrix}}^t = \begin{bmatrix}4&-3\\-5&4\end{bmatrix} $$ Then we plug it into the formula for the change of basis: $$B=Q^{-1}AP: \begin{bmatrix}4&-3\\-5&4\end{bmatrix}. \begin{bmatrix}1&2&3\\-1&1&-1\end{bmatrix}\quad.\begin{bmatrix}-1&3&-1\\-1&-1&0\\-2&1&1\end{bmatrix}\quad = \begin{bmatrix}-40&41&8\\31&-32&-6\end{bmatrix}\quad $$

so our matrix is $$B= \begin{bmatrix}-40&41&8\\31&-32&-6\end{bmatrix}\quad$$

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    Welcome to SE! Take some time to look around and explore to see how to ask good questions. You will find that your question will be answered much quicker and more thoroughly if it is asked in a high quality manner. Some tips are: keep you question self contained as opposed to pasting links. If you are going to post links, make sure they aren't broken (like your first one). There's a good place to start (y) – 2017-01-03
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    thank you a lot, and yeah sry mate, I didn't know it was broken. I will see what I can do to makei t better! – 2017-01-03
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    Heyo, check out this tutorial on MathJax for help with formatting: http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 2017-01-03
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    hello, yeah I was doing that, just tested it out! thank you for your help – 2017-01-03
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    I'm editing your question to show how to write the matrices in your image. It looks like there might be a typo in P because the middle element of the first column is $-1$ while the corresponding entry of $u_1$ is $1$. – 2017-01-03
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    @tonytouch Also what do you mean by "connection"? $A$ is the matrix representing the transform, and $P$ and $Q$ are the change of base matrices. – 2017-01-03
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    @MichaelBurr That is almost certainly a typo. – 2017-01-03
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    Thank you all so much! In my exercise it said: "show the connection between A,P,Q and matrix B" and then I translated it from my native language. I didn't say anything about the change of basis. – 2017-01-03
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    I've translated your first image into the question. Perhaps you can copy your second image into the question using my edits as a guide. – 2017-01-03
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    Yes, I will do that now, thank you very much! I tried to upvote all your comments, but I don't think I can atm, will try again later! :) – 2017-01-03
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    There’s no particular connection among the three matrices that you’ve shown. However, you also mention a matrix $B$ at the beginning of the problem. What is that? Is $B$ meant to be the matrix of $f$ relative to the given bases? The question would make sense to me then. – 2017-01-03
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    I just added matrix B. Matrix B represent f: $F^3 -> F^2$ relative to the basis $(u_1, u_2, u,3)$ for the domain and the basis $(v_1,v_2)$ for the co-domain. – 2017-01-03

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