How to change from a vector parametric description to a cartesian when describing a 2 dimensional flat in 5 dimensions?

Question

I'm completely stuck on how to start this question; I'm not sure If I can get one cartesian to describe this flat or if I need multiple?

Answer 1

We are in a $5$-dimensional vector space $V$ with two free parameters, so our vector subspace $U$ has dimension $2$.
If we now want to obtain its cartesian representation we need to find the scalar parameters $s$ and $t$ in function of the variables $v,w,x,y,z$ in order to obtain a system of three equations that describes a plane in the vector space $V$. $$\begin{cases} v = 1+s \\ w= 2+s \\ x =3\\ y=4+t\\ z=5 +t \end{cases} \iff \begin{cases} s = v-1 \\ w= 2+s \\ x =3\\ t=y-4\\ z=5 +t \end{cases} \iff \begin{cases} s = v-1 \\ t= y-4 \\ x =3\\ z=5+y-4\\ w=2+v-1 \end{cases} \Longrightarrow \pi\equiv\begin{cases} x-3=0 \\ y-z+1=0 \\ v-w+1=0 \end{cases}$$