1. ## dimensions and basis

M =
{( a b)
{(-b c) :a,b,c real numbers }

N =
{( x 0)
{(y 0) :x,y real numbers }

are subspaces of M2(R).

I need to find the dimension of M and N. how do i do this
is the dimension the number of elements in the basis?
if so how do i find a basis for these?
thank you

2. The basis for M is:
$\displaystyle \left\{ {\left[ {\begin{array}{rl} 1 & 0 \\ 0 & 0 \\ \end{array}} \right],\left[ {\begin{array}{rl} 0 & 1 \\ { - 1} & 0 \\ \end{array}} \right],\left[ {\begin{array}{rl} 0 & 0 \\ 0 & 1 \\ \end{array}} \right]} \right\}.$

3. yeah i thought it was but i worked it out just by looking at it. if im doing a proof is there a way i can devise a basis or does it have to satisfy anything to be a basis?
this means the dimension of M = 3?
what is M + N?

4. The sum M+N produces $\displaystyle M_2 (R)$.
To see this consider:
$\displaystyle \left[ {\begin{array}{rr} x & y \\ w & z \\ \end{array}} \right] = x\left[ {\begin{array}{rr} 1 & 0 \\ 0 & 0 \\ \end{array}} \right] + y\left[ {\begin{array}{rr} 0 & 1 \\ { - 1} & 0 \\ \end{array}} \right] + z\left[ {\begin{array}{rr} 0 & 0 \\ 0 & 1 \\ \end{array}} \right] + \left( {y + w} \right)\left[ {\begin{array}{rr} 0 & 0 \\ 1 & 0 \\ \end{array}} \right].$