# Prove that $W_1$ is a subspace of $\mathbb{R}^n$.

Let $$W_1=\{(a_1,a_2,...,a_n) \in \mathbb{R}^n : a_1 + a_2 + ... + a_n = 0 \}$$. We want to show that $$W_1$$ is a subspace of $$\mathbb{R}^n$$.

My attempt:

In order to show that $$W_1$$ is a subspace of $$\mathbb{R}^n$$, we must satisfy the following conditions: $$W_1 \subset \mathbb{R}^n$$, $$\exists 0 \in W_1$$, $$W_1$$ is closed under addition, and $$W_1$$ is closed under scalar multiplication. $$W_1$$ is clearly a subset of $$\mathbb{R}^n$$ because any element in $$\mathbb{R}^n$$ can be defined by a vector $$[a_1, a_2, ..., a_n].$$ Next, there exists a zero element in $$W_1$$ as $$a_1=a_2=...=a_n=0,$$ satisfies our criteria, namely $$(0,0,0,...,0)$$ is contained in $$W_1$$. To show closure under addition, take $$(a_1, a_2, ..., a_n), (b_1, b_2, ..., b_n) \in W_1$$, then $$(a_1, a_2, ..., a_n) + (b_1, b_2, ..., b_n) = (a_1+b_1, a_2+b_2, ..., a_n+b_n)$$ is also contained within $$W_1$$. Lastly, checking scalar multiplication, it must be that $$c(a_1, a_2, ..., a_n)=(ca_1, ca_2, ..., ca_n) \in W_1$$. Hence, $$W_1$$ is a subspace of $$\mathbb{R}^n$$.

I feel like I am taking great leaps of faith here, so I just want to double check that I am doing this correctly and understand the Subspace Criterion properly. Thank you for any pointers.

• It’s not enough for there to be some element of $W_1$ that acts as an additive identity. It must be identical to the zero element of the enclosing space. – amd Oct 3 '18 at 18:37

You just claimed, without any hint of a proof, that if $$(a_1, a_2,\ldots, a_n), (b_1, b_2,\ldots, b_n) \in W_1$$ and $$c\in\mathbb R$$, then $$(a_1, a_2,\ldots, a_n)+(b_1, b_2,\ldots, b_n) \in W_1$$ and $$c(a_1, a_2,\ldots, a_n)\in W_1$$. It's not hard, though. If $$(a_1, a_2,\ldots, a_n), (b_1, b_2,\ldots, b_n) \in W_1$$, then $$a_1+\cdots+a_n=b_1+\cdots+b_n=0$$. But then $$(a_1, a_2,\ldots, a_n)+(b_1, b_2,\ldots, b_n) \in W_1$$, since$$(a_1+b_1)+\cdots+(a_n+b_n)=a_1+\cdots+a_n+b_1+\cdots+b_n=0+0=0.$$Can you prove now that $$c(a_1, a_2,\ldots, a_n)\in W_1$$?