Hi.

i have the following subspaces:

$\displaystyle W=Span\left \{ (0,2,1), (1,2,0) \right \}$

$\displaystyle U=Span\left \{ (5,4,-5),(0,1,0) \right \}$

i need to find the basis for $\displaystyle U+W$ and $\displaystyle U\cap W$ and to determine if $\displaystyle U+W$ is a direct sum of $\displaystyle \mathbb{R}$.

so, this is what i did to find the basis for $\displaystyle U+W$:

i set $\displaystyle (5,4,-5),(0,1,0),(0,2,1), (1,2,0)$ in one $\displaystyle 3\times 4$ matrix and found its rank to be 3.

so i can just throw one of this vectors out, and it can be basis for $\displaystyle U+W$.

and obviously $\displaystyle U+W$ can't be direct sum of $\displaystyle \mathbb{R}$, since $\displaystyle U+W\neq \left \{ \vec{0} \right \}$ because $\displaystyle dim(U\cap W)=1>0$.

that all leaves me with 2 questions:

1. first of all: how do i find the basis for $\displaystyle U\cap W$?

2. is it enough to say that "$\displaystyle U+W$ can't be direct sum of $\displaystyle \mathbb{R}$, since $\displaystyle U+W\neq \left \{ \vec{0} \right \}$ because $\displaystyle dim(U\cap W)=1>0$" in order to prove that $\displaystyle U+W$ is not a direct sum of $\displaystyle \mathbb{R}$?

thanks in advanced!