Mark44 said:
For ii, you need to take two arbitrary matrices, say M and N, that are in U, and show that M + N is in U. For the two matrices M and N, the only entries that are important are the ones on the diagonal. I suspect that's where the hint will be successful.
You made some unwarranted assumptions in your matrix in part ii; namely, that the entries on the diagonal were 2a, -a, and -a. If the entries on the diagonal for M are a, b, and c, what makes it so that M is in U?
Does this prove that U is a subspace of V?:
{\bf 0}\;\text{matrix} \in U so U\neq\emptyset .
For For a,a',b,b',c,c',d,d',e,e',f,f',g,g',h,h',i,i'\in\mathbb{R} ,
Let A=\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\in U
and B=\begin{bmatrix} a' & b' & c' \\ d' & e' & f' \\ g' & h' & i' \end{bmatrix}\in U
so \text{trace}(A) = a+e+i=0 and \text{trace}({B}) = a'+e'+i'=0
\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} + \begin{bmatrix} a' & b' & c' \\ d' & e' & f' \\ g' & h' & i' \end{bmatrix} = \begin{bmatrix} a+a' & b+b' & c+c' \\ d+d' & e+e' & f+f' \\ g+g' & h+h' & i+i' \end{bmatrix}\in U
since (a+a')+(e+e')+(i+i')=(a+e+i)+(a'+e'+i')= \text{trace}(A) + \text{trace}(B) = 0
\therefore U is closed under vector addition.
\alpha A=\begin{bmatrix} \alpha a & \alpha b & \alpha c \\ \alpha d & \alpha e & \alpha f \\ \alpha g & \alpha h & \alpha i \end{bmatrix}\in U
since \text{trace}(\alpha A}) = \alpha a +\alpha e +\alpha i = \alpha (a+e+i) = \alpha \text{trace}(A) = 0
\therefore U is closed under scalar multiplication.
\therefore U is a subspace of V=M_3(\mathbb{R})