<s>[INST] a) Find a base for $W_1 \cap W_2$ and b) Find a base for $W_1 + W_2$ Let 
$$W_1 = \left\{ \begin{pmatrix}
    a & b & c \\
   -b & a & b \\
    c & b & a \\
\end{pmatrix} :\, a, b, c \in \Bbb R \right\}$$
$$W_2 = \left\{
    \begin{pmatrix}
    a & 0 & 0 \\
    b & 0 & c \\
    c & b & a \\
\end{pmatrix} :\, a, b, c \in \Bbb R \right\}$$
a) Find a basis for $W_1 \cap W_2$
b) Find a basis for $W_1 + W_2$
For a) I`m  not sure if the set for $W_1 \cap W_2 = $ $ \{
    \begin{pmatrix}
    0 & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 0 \\
    \end{pmatrix}\} 
$ 
so the base is only the zero matrix.
For b) I belive the set is: $W_1 + W_2 = $ $ \{
    \begin{pmatrix}
    2a & b & c \\
    0 & a & b+c \\
    2c & 2b & 2a \\
    \end{pmatrix}
$:$ a, b, c \in R\}$
But don`t know how to give an 
appropriate base for this set. 
 [/INST] For b), rewrite what you obtained as
$$a  \begin{pmatrix}
    2& 0& 0\\
    0 & 1 & 0 \\
     0 & 0& 2 \\
    \end{pmatrix}+b  \begin{pmatrix}
    0 & 1 & 0\\
    0 & 0 & 1 \\
    0 & 2  & 0 \\
    \end{pmatrix}+c  \begin{pmatrix}
   0 & 0 &1\\
    0 & 0 & 1 \\
    2  & 0 & 0 \\
    \end{pmatrix}$$
Can you say whether the three above matrices are linearly independent?
</s>