Question

Suppose U and V are subspace of R^n. Prove that orthogonal of ( U intersection V)=orthogonal of U+ orthogonal of V

Answer 1

for $u,v\isin U\cap V$ :

if $x\isin (U\cap V)^{\perp}$ and $U\cap V \neq 0$ then:

$x\cdot u=0$ or $x\cdot v=0$ , so

$x\isin U^{\perp}+V^{\perp}$

so,

$(U\cap V)^{\perp}$ is subset of $U^{\perp}+V^{\perp}$

if $x\isin U^{\perp}+V^{\perp}$ then:

$x\cdot u=0$ or $x\cdot v=0$ for $u\isin U$ and $v\isin V$

then $x\isin (U\cap V)^{\perp}$

so,

$U^{\perp}+V^{\perp}$ is subset of $(U\cap V)^{\perp}$

that is, $(U\cap V)^{\perp} =U^{\perp}+V^{\perp}$