Question

Determine if the subset of R^4 is a subspace. {(x1,x2,x3,x4)|x1+2(x2)=0}

Answer 1

from wikipedia
Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:
The zero vector, 0, is in W.
If u and v are elements of W, then any linear combination of u and v is an element of W;
If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

Now, I'm not familiar with the notation you are using. But I don't see the zero vector.

Answer 2

\(W=\) {\((x_1,x_2,x_3,x_4)|x_1+2x_2=0\)}={\((-2x_2,x_2,x_3,x_4)|x_1,x_2,x_3,x_4 \in R\)}

we must show that every \(\vec v=(-2a,a,b,c) \in W\) is closed under addition and scalar multiplication

(I) closed under scalar multiplication

let \(d \in R\) be a real number then \[d \ \vec v=(-2ad,ad,bd,cd)=(-2k,k,m,n) \in W\]

now u do the same for addition