Subspace of vectors in $R^4$

Question

Question: If $W= \left \lbrace [x_1, x_2, x_3, x_4]\in R^4 : x_1 = x_3 - x_4, x_2 = x_3 + x_4 \right \rbrace,$ determine if $W$ is a subspace of $R^4.$

Solution:

(i) Let $x_3 = 1, x_4 = 2 \implies x_1 = -1, x_2 = 3$

Since $[-1,3,1,2] \in W$, $W$ is nonempty

(ii) If vectors $x = [x_1, x_2, x_3, x_4], y = [y_1, y_2, y_3, y_4] \in W$

$$x+y = [x_1 + y_1, x_2 + y_2, x_3 + y_3, x_4 + y_4] = ??$$

Don't really know if I should expand it completely to show that it closes under vector addition

(iii) do I just do $r[x_1, x_2, x_3, x_4] = [rx_1, rx_2, rx_3, rx_4]$ to show it closes until scalar multiplication for $r \in R$?

Answer 1

Note that you can write $W=\Big\{(x_1, x_2, x_3, x_4)\in \mathbb{R}^4 \Big|x_1 = x_3 - x_4, x_2 = x_3 + x_4\Big\}$ as $W=\Big\{(x-y,\,x+y,\,x,\,y)\in \mathbb{R}^4\,\Big|\,x,y\in \mathbb{R}\Big\}$

Now consider $x_!,x_2,y_1,y_2 \in \mathbb{R}$. Then

$(i)$ $(0,0,0,0) \in W$ $($Putting $x=0,y=0$$)$

$(ii)$ $c\,(x_1-y_1,\,x_1+y_1,\,x_1,\,y_1)=(cx_1-cy_1,\,cx_1+cy_1,\,cx_1,\,cy_1)\in W$

$($Putting $x=cx_1,\,y=cy_1$$)$

$(iii)$ $(x_1-y_1,\,x_1+y_1,\,x_1,\,y_1)+(x_2-y_2,\,x_2+y_2,\,x_2,\,y_2)=((x_1+x_2)-(y_1+y_2),\,(x_1+x_2)+(y_1+y_2),\,x_1+x_2,\,y_1+y_2)\in W$

$($Putting $x=x_1+x_2,\,y=y_1+y_2$$)$

Note that $(i)$ essentially follows from $(ii)$ by taking $c=0$. But we check that as a necessary condition. If that condition fails the solution ends then and there.