Let $u$ the subspace generated by the vector $[2,1]$ and $w$ the subspace generated by the vector $[0,1]$
Show that $\mathbb{R}^2$ is a direct sum of subspaces $u,w$
I have already proven that the intersection contains $0$
But now How i prove that every vector in $\mathbb{R^2}$ can be written as sum of unique vectors in $u$ and $w$ I try assuming that exists two differents factorization of $x\in\mathbb{R^2}$
and then
$(x,y) = a(2,1)+b(0,1)$ and
$(x,y) = \lambda(2,1)+\mu(0,1)$
then $x=2a $ and $y=a+b$
and
$x=2\lambda$ and $y=\lambda +\mu$ so
$2\lambda=2a \rightarrow \lambda=a $ and $a+b=\lambda + \mu \rightarrow b=\mu$ because $a=\lambda$
and then exist a unique factorization for every vector in $\mathbb{R^2}$
isi right?