Finding basis of vector subspaces Let $$A=\left\{A\in \mathbb{R}^{3\times 3}\mid A^t=A \right\}$$ be the set of real symmetric $3\times 3$-matrices and let $$B=\left\{B\in \mathbb{R}^{3\times 3}\mid t^B=B^t\right\},$$ here $t^B$ is the transpose of $B$ w.r.t. the antidiagonal.
1.) Prove that $A$ and $B$ are subspaces.
2.)Find the bases for vector subspaces $A$,$B$,$A\cap B$, $A+B$.
For 1. I think I should just check if: $αA_1 + βA_2$ is still in $A$ (similarly for $B$). But I am unsure how to do the second part. I am asking the question for the first time, so I hope I wrote the problem correctly. I thank you for the response!
 A: 1) requires from you to simply follow definition. I use the one given in Wiki, you should use the one provided in your lectures/textbook (there exist slightly different ways of defining the same entity). For $A$, checking whether $A$ is a linear subspace of $\mathbb{R}^{3\times 3}$ (over field $\mathbb{R}$):
0. $A$ is a subset of $\mathbb{R}^{3\times 3}$. Obviously true, check.
1. Zero vector $0$ is in $A$. Zero vector in $\mathbb{R}^{3\times 3}$ is zero matrix, zero matrix is symmetric. Check.
2. $u, v \in A \implies u+v = w \in A$. Sum of symmetric matrices is symmetric (you can write rigorous proof). Check.
3. $u \in A, c \in \mathbb{R} \implies cu \in A$. Symmetric matrix multiplied by a scalar is still symmetric. Check.
Note that there may be shortcuts (in particular, you're right when saying that proving $A_1, A_2 \in A, \alpha, \beta \in \mathbb{R} \implies \alpha A_1+\beta A_2 \in A$ is sufficient), but unless you've grown really confident in the subject or a shortcut saves significant effort, I wouldn't recommend to use them.
2) requires you to find a basis in a certain subspace. Once again, start from definition, I'm using Wiki. First condition (non-existence of nontrivial linear combination equal to zero) is clearly equivalent to "any vector of basis can't be expressed as a linear combination of others" (why?). For example, set of nine matrices with one element $1$ and all other $0$ in each - 
$$\pmatrix{1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0}, \pmatrix{0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0}, ..., \pmatrix{0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1}$$
 would give you a basis of $\mathbb{R}^{3\times 3}$.
Bad news: there is no universal way to find a basis for a vector space, you always have to make some sort of guess. Taking $B$ as an example, I need $$\pmatrix {x_{1,1} & x_{1,2} & x_{1,3} \\ x_{2,1} & x_{2,2} & x_{2,3} \\ x_{3,1} & x_{3,2} & x_{3,3} } : \pmatrix {x_{1,1} & x_{2,1} & x_{3,1} \\ x_{1,2} & x_{2,2} & x_{3,2} \\ x_{1,3} & x_{2,3} & x_{3,3} } = \pmatrix {x_{3,3} & x_{2,3} & x_{1,3} \\ x_{3,2} & x_{2,2} & x_{1,2} \\ x_{3,1} & x_{2,1} & x_{1,1} }$$
This gives me four non-trivial equalities ($x_{1,1} = x_{3,3}, x_{2,1}=x_{2,3}, x_{3,1} = x_{1,3}, x_{1,2} = x_{3,2}$) and any matrix satisfying them is in $B$. In other words, I get to define five matrix elements (say, $x_{1,1} = \alpha, x_{2,1} = \beta, x_{3,1} = \gamma, x_{1,2} = \delta, x_{2,2} = \epsilon$) however I want and others can be derived. Or, rephrasing again, $$b \in B \iff b = \alpha \pmatrix{1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1} + \beta \pmatrix{0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0} + ... + \epsilon \pmatrix{0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0}$$
These five matrices (write down the missing two) seem like a basis. Double-checking: they can't give a non-trivial zero linear combination (each has $1$ in a position where others have $0$) and any element of $B$ can be produced as their linear combination. Done.
A: In 2. you got to create a Basis. This means, that you get a set of elements of that subspace, whose linear combinations can create all elements of the subspace, bot not a single element, that is not in the subspace. 
This is most easily done, by writing a generic $3 \times 3$ matrix done and identify, which elements have what relation to reduce the numbers of degrees of freedom. 
Example: Given the symmetric $3\times 3$, we know, that $x_{2,1}$ must be equal to $x_{1,2}$, so we can elimnate one of these from the basis. 
In a more formal (and algorithmic) manner you could use any basis you find and use orthogonlization-shemes (like Gram-Schmidt or similar). 
The question is also not clearly written, since there are infinite bases. It might want to ask for the simplest orthogonal basis, but that is not specific. 
