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I am having trouble understanding my problem and what to calculate

I have been given the subspace $U=\{x=$$ \begin{vmatrix} x_1\\ x_2\\ x_3\\ \end{vmatrix} \in F^3 | x_1 + x_2 + x_3 = 0\} \subset F^3 $

and the linear transformation $f: U \rightarrow F^2 $ $f\begin{vmatrix} x_1\\ x_2\\ x_3\\ \end{vmatrix} =\begin{vmatrix} x_1\\ x_2+x_3\\ \end{vmatrix}$

The question is to determine a matrix A that represent $f: U \rightarrow F^2 $ with respect to the basis for U and standard basis $(e_1,e_2)$ for $F^2$

My attempt: I have calculated the basis $B=\{\begin{vmatrix} 1\\ 0\\ -1\\ \end{vmatrix},\begin{vmatrix} 0\\ 1\\ -1\\ \end{vmatrix}\}$

And my matrix A calculated from the linear transformation

$\begin{vmatrix} 1&0&0\\ 0&1&1\\ \end{vmatrix}$

I'm aware the standard basis are $e_1=\begin{vmatrix} 1\\ 0\\ \end{vmatrix} , e_2=\begin{vmatrix} 0\\ 1\\ \end{vmatrix}$

I'm not sure about the next step. Do I calculate: $f\begin{vmatrix} 1\\ 0\\ -1\\ \end{vmatrix} =\begin{vmatrix} 1\\ -1\\ \end{vmatrix}$

and do the same thing for the other basis or am I suppose to find a matrix A that's going to give me $f\begin{vmatrix} 1\\ 0\\ -1\\ \end{vmatrix} =\begin{vmatrix} 1\\ 0\\ \end{vmatrix}$

and

$f\begin{vmatrix} 0\\ 1\\ -1\\ \end{vmatrix} =\begin{vmatrix} 0\\ 1\\ \end{vmatrix}$

I just don't understand the question. How am I suppose to find a matrix A with respect to the basis of U and the standard basis for $F^2$?

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The image of the first vector in the basis is $$ \begin{bmatrix} 1 \\ 0+(-1) \end{bmatrix} $$ and the image of the second vector is $$ \begin{bmatrix} 0 \\ 1+(-1) \end{bmatrix} $$ so the matrix is $$ \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} $$ Note that the map $f$ can be more easily described as $$ f\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_1 \\ -x_1 \end{bmatrix} $$

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  • $\begingroup$ Thats it? I'm afraid I don't fully understand how this is a representation of the standard basis despite it being simple and straightforward. So that means the matrix you have written is the representing matrix and the answer to the problem and not my matrix A calculated from the linear transformation? $\endgroup$ – mahma Jan 3 at 0:08
  • $\begingroup$ @mahma The map is $f\colon U\to F^2$ and the domain has dimension $2$, as well as the codomain; so the representing matrix must be $2\times2$. $\endgroup$ – egreg Jan 3 at 0:47

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