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**Jose27** What is $\displaystyle (\vec{ax},\vec{by},\vec{cz}),(\vec{ax'},\vec{by'}, \vec{cz'})\in{W}$?

To show that W is closed under addition and scalar multiplication take $\displaystyle w_1=a_1x + b_1y +c_1z$ and $\displaystyle w_2=a_2x+b_2y+c_2z$ where $\displaystyle a_i,b_i \in \mathbb{R}$ and $\displaystyle r \in \mathbb{R}$. Now $\displaystyle w_1+w_2=(a_1x + b_1y +c_1z)+(a_2x+b_2y+c_2z)=(a_1+a_2)x+(b_1+b_2)y+(c_1 +c_2)z$ and since $\displaystyle \mathbb{R}$ is closed under addition we get that $\displaystyle w_1+w_2=a_3x+b_3y+c_3z$ where $\displaystyle a_3=a_1+a_2$ and so on. In the same manner $\displaystyle rw_1=r(a_1x+b_1y+c_1z)=(ra_1)x+(rb_1)y+(rc_1)z$ and $\displaystyle \mathbb{R}$ is closed under multiplication...and the rest you can do.