Help Identifying Vector Spaces. 1) The set of all $2 \times 2$ matrices of the form $\begin{pmatrix}
a & b\\ 
c & -a
\end{pmatrix}$,
2) The set of all vectors to form $\begin{pmatrix}
a - 1\\
a + 1
\end{pmatrix}$,
3) The set of all even polynomials $p(x) = p(-x)$.
I know what a vector space is but I don't know how to prove it.
Thanks
I have to satisfy this 


*

*Commutativity: v + w = w + v for all v, w ∈ V ;

*Associativity: (u + v) + w = u + (v + w) for all u, v, w ∈ V ;

*Zero vector: there exists a special vector, denoted by 0 such that
v + 0 = v for all v ∈ V ;

*Additive inverse: For every vector v ∈ V there exists a vector w ∈ V
such that v + w = 0. Such additive inverse is usually denoted as −v;

*Multiplicative identity: 1v = v for all v ∈ V ;

*Multiplicative associativity: (αβ )v = α(β v) for all v ∈ V and all
scalars α, β ;

*α(u + v) = αu + αv for all u, v ∈ V and all scalars α;

*(α + β )v = αv + β v for all v ∈ V and all scalars α, β .
so lets say that v = $\begin{pmatrix}
a & b\\ 
c & -a
\end{pmatrix}$,
so what should i make w?
 A: To show that something is a vector space you need to check a few axioms. However, to check that something is a vector subspace (and thus itself a vector space) you only need to check a few things. Thus a subset $S\subseteq V$ of a vector space $V$ is a subspace of $V$ if $\{0\}\subseteq S $ and if $S$ is closed under addition of vectors and scalar multiplication. 
So, for each of the cases above, identify first a larger vector space where the given set lives. (e.g., the vector space of all $2\times 2$ matrices, the vector space of polynomials etc.). Then, check if the zero vector is in the subset you consider. For instance in 2) above, the zero vector is not in the subset so you can automatically rule it out as a subspace of the vector space $\mathbb R^2$. Then, check if the given subset is closed under addition of vectors and scalar multiplication. 
A: To take on the last part of the question, the matrices of the form specified are, in English, the matrices where the upper-left and lower-right entries add up to zero. So $w$ should be an arbitrary matrix meeting that specification, say, $\pmatrix{q&r\cr s&-q\cr}$, and then you have to check whether $v+w$ also meets the specification. 
