# How to calculate symmetry

This is a task from an old exam.

Let define:

$V = \text{lin}((1,2,2,1),(1,1,-1,1))$

$W = \cases{x_1-x_2=0\\x_3-x_4 =0}$

$\varphi:\mathbb{R}^{4}\to \mathbb{R}^{4}$ - a symmetrion against V along W.

Please help me calculate $\varphi((1,1,2,0))$

Let $a:=(1,2,2,1)$, $b:=(1,1,-1,1)$ for $V$ and $c:=(1,1,0,0)$ and $d:=(0,0,1,1)$ for $W$ -- you can check that $W=\mathrm{lin}(c,d)$.
So, we want to decompose the given vector with these: $-b+d=(-1,-1,2,0)$, so add $+2c$ and we're there.
I didn't exactly understand this 'symmetrion against $V$', I guess the directions in $V$ would be reversed and those of $W$ would stay (or the other way around). So, apply $\varphi$ accordingly to $b,c,d$ and use its linearity.