Answer to Question #282359 in Linear Algebra for Rizwana

Question #282359

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

1
Expert's answer
2021-12-26T16:30:32-0500

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

if x(UV)x\isin (U\cap V)^{\perp} and UV0U\cap V \neq 0 then:

xu=0x\cdot u=0 or xv=0x\cdot v=0 , so

xU+Vx\isin U^{\perp}+V^{\perp}

so,

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


if xU+Vx\isin U^{\perp}+V^{\perp} then:

xu=0x\cdot u=0 or xv=0x\cdot v=0 for uUu\isin U and vVv\isin V

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

so,

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


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


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