If $x,y \in V$ are linearely independent, then there exists a transvection $\tau$ with $\tau(x)=y$

Question

Let $V$ be a $n$-dimensional $K$ vector space. A $\tau \in \operatorname{GL}(V)$ is called a transvection, if there exists a $(n-1)$-dimensional $\tau$-invariant subvector space $W$ of $V$ with

$$\tau_{W} = id_W \text{ and } \tau(v)-v \in W \quad \forall v \in W$$

Task:

Prove: If $x,y \in V$ are linearly independent, then there exists a transvection $\tau$ with $\tau(x)=y$.

All the mappings of $\tau$ that I can find either do not satisfy $\tau(x)=y$ or $\tau(v)-v \in W $.

What I have so far:

$x,y$ are linearly independent $\Rightarrow \operatorname{V} \geq 2$ $\Rightarrow \text{because} W=<y> \Rightarrow \operatorname{dim}(V) > dim(W) > 0$ $\Rightarrow \text{so } W \text{exists and thus } \tau$.

I have found a similiar post on that task. But it seems not to be the answer to my task. If yes, please help me clarifying that answer.

Answer 1

Since $\{ x, y \}$ are linearly independent, we also have that $\{ x - y, x \}$ are linearly independent. Complete this set to a basis

$$ w_1 = x - y, w_2, \dots, w_{n-1}, w_n = x $$ for $V$ and set $W = \operatorname{span} \{ w_1, \dots, w_{n-1} \}$. Define a linear map $\tau \colon V \rightarrow V$ by requiring that $$ \tau(w_i) = w_i, \,\,\, 1 \leq i \leq n - 1, \\ \tau(w_n) = y. $$

Then clearly $\tau|_{W} = \operatorname{id}|_{W}$ and $\tau(w_n) = \tau(x) = y$. Finally, any $v \in V$ can be written as $v = w + ax$ for some $w \in W$ and $a \in K$ and then

$$ \tau(v) - v = \tau(w + ax) - (w + ax) = \tau(w) + a\tau(x) - w - ax = w + ay - w - ax = (-a)(x - y) \in W. $$

Answer 2

Choose a basis where the first to vectors are $x$ and $y$ then the transformation $\tau$ corresponding to the cyclic permutation matrix of order $2$: $$ \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ \end{pmatrix} $$ will do the trick. Note that $W$ is the kernel of $\tau - \tau^0$ which has rank $1$ since $ \begin{pmatrix} -1 & 1 \\ 1 & -1 \\ \end{pmatrix} $ has rank $1$, so dim($W$) = $n-1$.