Decide if the following statements are true or false. Provide arguments or a couterexample to support your answers:
1) The set of matrices $A$ with $\det(A)=1$ is a subspace in the vector space $\mathcal{M}_{2 \times 2}(\mathbb{R})$ of $2 \times 2$ matrices.
2) $\dim(\operatorname{Null}(A))=\dim(\operatorname{Null}(S_{A})).$
3) If $S=\operatorname{span}(u_{1},u_{2}, \ldots, u_{n})$ then $\dim(S)=n$.
4) The intersection of two vector subspaces of a vector space $V$ cannot be empty.
5) In the vector space $\mathcal{M}_{2 \times 2}(\mathbb{R})$ consider $M$, the set of matrices with positive elements. The subspace spanned by the matrices from $M$ is $\mathcal{M}_{2 \times 2}(\mathbb{R})$ itself.
I think the affirmative answers are for the questions 1), 3). But about the remaining questions I can't tell anything. I am not sure, but I think 4) is affirmative, also--but I'm not sure.
Thanks :)