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Let $W = \langle(1,-1,2,1)\rangle$ and $V = \{(x,y,z,t)\ |\ x+y-z-t=0\}$ be subspaces of $\mathbb{R}^4$. What is the basis for $V+W$ and $V\cap W$?

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Everything in $W$ is of the form $(a,-a,2a,a)$. But $$ a+(-a)-2a-a=0 $$ only when $a=0$. Thus $V\cap W=\{0\}$. Thus a basis for $V+W$ is just a basis for $V$ along with a basis for $W$.

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And the basis for $V\cap W$ is $\emptyset$, in case it wasn't obvious from $V\cap W=\{\emptyset\}$. – Mario Carneiro Dec 15 '12 at 11:03

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