Finding Orthogonal Complement

Video

https://invidious.yoshixi.net/watch?v=A5MHWONSqRw

Example

WIth $U=span\{(3,-1,0),(1,0,1)\}$ Find the orthogonal complement $U^{\perp }$

• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|⟨v|u⟩=0,\forall u\in U\}$
• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|⟨v|span\{(3,-1,0),(1,0,1)\}⟩=0\}$
• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|⟨v|a(3,-1,0)+b(1,0,1)⟩=0\}$
• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|a⟨v|(3,-1,0)⟩+b⟨v|(1,0,1)⟩=0\}$
• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|span\{⟨v|(3,-1,0)⟩,⟨v|(1,0,1)⟩\}=\{0\}\}$
• Note that the only time the span of two real numbers is 0, is if both numbers is 0
• Then, $U^{\perp }=\{v\in {\mathbb{R}}^3|⟨v|(3,-1,0)⟩=⟨v|(1,0,1)⟩=0\}$
• With $v=(x,y,z)$, We get from solving the equations :
  • $y=3x$
  • $z=-x$