# is this subset a subspace - redux

OK, I have been bothering people here with this for days and with luck I finally have this. People have helped a lot here so far. (Doing these examples is I hope helping me learn the proofs, but I want to know that I am doing this right).

Let W be a subset of vector space V. Is it s subspace as well?

W = {($a_1$, $a_2$, $a_3$) ∈ $ℝ^3$ : $2a_1 - 7a_2 + a_3=0$}

So, to check if this is a subspace I need to satisfy the following:

1. That 0 is in the set. Plugging (0,0,0) into the equation $2a_1 - 7a_2 + a_3=0$ yields 0=0 so yes, it is.

2. That it is closed under addition.

Let ($b_1, b_2, b_3$) be an arbitrary vector in W.

For this to be closed under addition ($b_1, b_2, b_3$)+($a_1, a_2, a_3$) ∈ W.

$2(a_1+b_1) - 7(a_2+b_2) + (a_3+b_3) = 0$

can also be written as $(a_3+b_3) = -2(a_1+b_1) + 7(a_2+b_2)$

There are real-valued solutions to this, whenever $b_i = -a_i$ is one, so the answer is yes, it is closed under addition.

1. Is it closed under multiplication?

Any arbitrary λ($2a_1 - 7a_2 + a_3)=0 =(λ)0$

So since that's still part of the set, it is closed under multiplication.

So, did I do this one correctly? God I hope so.

Excellent work. It will get easier! Trust me.

Very nicely done. You "covered all the bases", a bit awkwardly but you got the job done!

I'd just add: "Therefore, (since ${\bf 0} \in W$, and $W$ is closed under vector addition and scalar multiplication,) $\;W\,$ is a subspace of $\,V$.

This is how I'd approach the "closed under addition" component of the proof.

Let $w_1 = (a_1, a_2, a_3), w_2 = (b_1, b_2, b_3) \in W$.

So $2a_1 - 7 a_2 + a_3 = 0, \text{ and}\; 2b_1 - 7b_2 + b_3 = 0.$

Now, it follows that $w_1 + w_2 = (a_1 + b_1, a_2 + b_2, a_3 + b_3).$

And since we have $$2(a_1+b_1) - 7(a_2+b_2) + (a_3+b_3)$$ $$= 2a_1 + 2b_1 -7a_2 -7b_2 + a_3 + b_3$$ $$= (2a_1-7a_2+a_3)+(2b_1-7b_2+b_3)$$ $$={\bf 0 + 0} = {\bf 0}\in W$$

...$W$ is closed under vector addition.

• thanks, I am trying to work through these examples and make sure I understand what I am doing. Unlike vector calc it's much more abstract and I am struggling with that. – Jesse Jun 17 '13 at 3:42
• It will get easier, trust me. I mean that. – Namaste Jun 17 '13 at 3:44
• Nicely done Amy. Complete! +1 – mrs Jun 17 '13 at 3:52

I'm not sure I understand your proof that it is closed under addition (specifically, the $b_i=-a_i$ part seems like a special case, and not a proof). I recommend instead that you note: \begin{align}2(a_1+b_1)-7(a_2+b_2)+(a_3+b_3) &= 2a_1+2b_1-7a_2-7b_2+a_3+b_3\\ &= (2a_1-7a_2+a_3)+(2b_1-7b_2+b_3)\\ &= \textbf{0}+\textbf{0}\\ &= \textbf{0}.\end{align}