Determining whether a set of vectors is a subspace of a vector space I have a textbook question which asks to determine which of the following are subspaces of $R^3$:
(a) All vectors of the form (a,0,0)
(b) All vectors of the form (a,1,1)
From what I understand, I have to show that if two vectors $\vec u$ and $\vec v$ are an element of $R^3$ then $\vec u$ + $\vec v$ and k$\vec u$ are also an element of $R^3$. What I don't understand is how you are meant to show that $\vec u$ + $\vec v$ and k$\vec u$ are still elements of the vector space. 
Any help will be appreciated!
 A: (a) is a subspace because for any $(p,0,0)$ and $(q,0,0)$
$$(p,0,0)+(q,0,0)=(p+q,0,0)$$
which is still in the form of $(a,0,0)$
So is
$$k(p,0,0)=(kp,0,0)$$
(b) is not a subspace because for any $(p,1,1)$ and $(q,1,1)$
$$(p,1,1)+(q,1,1)=(p+q,2,2)$$
which is not in the form of $(a,1,1)$.
$$k(p,1,1)=(kp,k,k)$$
is also not in the form of $(a,1,1)$ for general $k$.
In fact, you can tell immediately that (b) is not a subspace, because it doesn't contain the null vector (0,0,0).
A: Two vectors of the form  $(a,0,0)$, say $(a_1,0,0) $ and $(a_2,0,0) $ have sum $(a_1+a_2,0,0) $ which is of the form  $(a,0,0) $.  As for scalar multiplication we have $c \cdot  (a,0,0)=(ca,0,0) $ is again of the form $ (a,0,0) $...
I will leave  (b) for you. ..
A: For proving $Y$ is a subspace of a vector space $X$,we must show that for any $a,b \in Y$,$a+b \in Y$ and $k.a \in Y$ for $k \in \Bbb{F}$ where $\Bbb{F}$ is the field associated with the Vector space.
So inyour above try you should consider elements from subspace rather than from $\Bbb{R}^{3}$.
In the first bit let $A = \{(a,0,0) |a \in \Bbb{R} \}$,so let's take any two elements from set $A$ say, $a = (a_{1},0,0) , b = (a_{2},0,0)$ then $a+b = (a_{1}+a_{2},0,0) \in A$ 
But this doesnot happen in 2nd bit as $B = \{ (a,1,1) |a \in \Bbb{R} \}$,let $u = (u_{1},1,1) , v = (u_{2},1,1)$ then $u+v = (u_{1}+v_{1},2,2) \notin B$,so the second one is not a subspace of $\Bbb{R}^{3}$ 
