> 
> Consider the set of matrices $2 \times 2$ whose dash is zero, i.e.
> U = {$A ​​\in M\_{2 \times 2}(\mathbb R) \vert tr(A) = 0$}.Show that $U$ is a subspace of $M\_{2 \times 2}(\mathbb R)$ and find a basis for $U$.
> 
> 
> 

My attempt

$tr(A) = 0 = a\_{11}+a\_{22}$
$A\_{2 \times 2} + B\_{2 \times 2} = [a\_{11}+b\_{11}, a\_{12}+b\_{12}, a\_{21}+b\_{21}, a\_{22}+b\_{22}] = [a\_{11}+b\_{11}, a\_{12}+b\_{12}, a\_{21}+b\_{21}, (-a\_{11}-b\_{11})] \to a\_{11}+b\_{11}+ (-a\_{11}-b\_{11}) = 0$

$\alpha A = [\alpha a\_{11}, \alpha a\_{12}, \alpha a\_{21}, \alpha a\_{22}] \to \alpha (a\_{11} + a\_{22}) = \alpha 0 = 0$

b={[1, 0, 0, 0], [0,1,0,0],[0,0,1,0],[0,0,0,-1]}

linear independence

$\alpha [1, 0, 0, 0], \beta [0,1,0,0], \gamma [0,0,1,0], \omega [0,0,0,-1] = [\alpha, \beta, \gamma, \omega] = [0,0,0,0] \to \alpha = \beta = \gamma \omega = 0$

$[a,b,c,-a] = a [1, 0, 0, 0], b [0,1,0,0], c [0,0,1,0], d [0,0,0,-1] = [a,b,c,-a]$

Correct?

Thanks.