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 🙂
Best Answer
1) The matrices having determinant equal to 1 form a group called special linear group. They do not form a vector subspace of $M_{2,2}$, since zero matrix does not belong to $\operatorname{SL}(2,F)$.
3) What if $u_1=\dots=u_n$? This is true only if the vectors are linearly independent.
5) Notice that this subspace contains matrices $A_1=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, $A_2=\begin{pmatrix}1&2\\1&1\end{pmatrix}$, $A_3=\begin{pmatrix}1&1\\2&1\end{pmatrix}$, $A_4=\begin{pmatrix}1&1\\1&2\end{pmatrix}$ and $A_5=\begin{pmatrix}1&1\\1&1\end{pmatrix}$. Can you obtain matrices from the standard basis as their linear combinations?
The part 4 was solved in comments. Without knowing what $S_A$ means, I can't really say anything about the part 2.