Give an explicit isomorphism between two sets

Question

Give an explicit isomorphism between the set of solutions of the linear system (over $\Bbb R$)

$$w − x + 0y + 3z = 0$$

$$w − x + y + 5z = 0$$

$$2w − 2x − y + 4z = 0$$

such problem has never been covered in our textbook, anyone has any ideas?

Answer 1

I assume you are looking for bijections between the three solution sets of the equations (one for each equation).

Let $S_1,S_2,S_3$ be the three solution sets.

For all $i=1,2,3$ and each choice of $(w,x,y)\in \mathbb{R}^3$ there is a unique $z\in \mathbb{R}$ such that $(w,x,y,z)\in S_i$.

So let $f:S_1 \rightarrow S_2$ be defined by $f((w,x,y,z))\mapsto (w,x,y,\frac{-w+x-y}{5})$

and let $g:S_2 \rightarrow S_3$ be defined by $f((w,x,y,z))\mapsto (w,x,y,\frac{-2w+2x+y}{4})$

and let $h:S_3 \rightarrow S_1$ be defined by $f((w,x,y,z))\mapsto (w,x,y,\frac{-w+x}{3})$