Suppose V= { v = ( $v_ 1$ ,$v_ 2$ ,$v_ 3$ $) ^T$ $\in$ $\mathbb{R}$ $^3$ | $v_ 2$ = 0 } and W$ = \mathbb{R} ^2$ . Furthermore ( A ,V,φ ) = ( V,V,φ ) and ( B ,W,ψ ) = ( W,W,ψ ) the respective affine Spaces. The trace mapping is defined between the respective spaces is then defined as:
$Tr f$ $\\$ : V → W,
v → $( v_ 1, v _3 )^T$ $\\$
Determine f.
I am completely confused by this question but the exact formulation is the one I wrote above. First of all, I don't even know how a trace mapping can have two dimensions. Furthermore it is unclear whether $f$ is a linear mapping, but maybe I am overlooking something. Can someone shed some light on this mistery of a question ?
It seems that the simpler interpretation of the question if to find the matrix of the transformation: $$ (\nu_1,0,\nu_3)^T\to(\nu_1,\nu_3)^T $$ and this is $$ \begin{pmatrix} 1&a&0\\ 0&b&1 \end{pmatrix} $$ with $a,b \in \mathbb{R}$. Clearly the term ''trace'' here is not the usual trace of the matrix.