Identity map with change of basis Let a map with the field $K$, from $f:K^3 \rightarrow K^2$ be defined as $f\left (\begin{bmatrix}
x_1\\ 
x_2\\ 
x_3
\end{bmatrix}  \right )=\begin{pmatrix}
x_1+2x_2+3x_3\\ 
3x_1+2x_2+x_3
\end{pmatrix}$
and the basis $\begin{pmatrix}
-1\\ 
0\\ 
2
\end{pmatrix}, \begin{pmatrix}
3\\ 
1\\ 
-1
\end{pmatrix}, \begin{pmatrix}
1\\ 
-1\\ 
1
\end{pmatrix}$ for $K^3$ and the standard basis $(e_1,e_2,e_3)$ for the codomain
Question
Find the matrix that represents $Id:K^3 \rightarrow K^3$ with respect to the given basis for the domain and the standard basis $(e_1,e_2,e_3)$ codomain.
I have found the matrix that represents $f:K^3 \rightarrow K^2$ with respect to the given standardbasis for the domain and the codomain but this identity map is really difficult for me to get.
Hint: Given any linear transformation, the columns of the matrix that represents that map are exactly the images of the basis vectors of the domain, expressed in the basis of the codomain.
Sorry, but the question as you asked it doesn't make any sense. I'm pretty sure you mixed together two unrelated questions. I suggest you check the source (the textbook or the instructor's original assignment), and I bet these are two separate parts being two separate questions, not a single question.
For starters, if the given function is $f:K^3\to K^2$, then we can't have &quot;the standard basis $(e_1,e_2,e_3)$ for the codomain&quot;, because the dimension of the codomain is $2$, not $3$.
My guess is that there's a separate question about finding the matrix representing the given transformation $f$ with respect to some bases in the domain $K^3$ and in the codomain $K^2$, and there's a separate question about finding the matrix representing the identity map $\operatorname{Id}:K^3\to K^3$ with respect to the given basis in the domain $K^3$ and the standard basis in the codomain $K^3$ — but the map $f$ has nothing to do with this question.
To address this latter question, the answer is already in the question. :-) Technically speaking, you need to do what other responses have already said. To find the matrix of a linear transformation $T$, such as e.g. $T=\operatorname{Id}$, you do the following: for each given basis vector $\mathbf{b}_i$ (of the &quot;domain&quot; basis), calculate $T(\mathbf{b}_i)=\operatorname{Id}(\mathbf{b}_i)$, and the components of the resulting vectors with respect to the codomain basis will be the corresponding columns of the desired matrix. And since in the codomain we're using the standard basis, what are we going to see as these columns?