left / right perpendicular

Given a sesquilinear form $b:V\\times V\\rightarrow k$ over the field $k$ , if $v,u\\in V$ such that $b(v,u)=0$ then we say $v$ is right perpendicular to $u$ and denote it $v\\bot u$ . Likewise $u$ is left perpendicular to $v$ and can be denoted by $u\\top v$ .

By definition $v\\perp u$ if and only if $u\\top v$ . However, $v\\perp u$ need not imply $u\\perp v$ .

For example, let $V=k\\oplus k$ and $b((v_{1},v_{2}),(u_{1},u_{2}))=v_{1}u_{2}$ . Then $b((0,1),(1,0))=0$ so $(0,1)\\bot(1,0)$ , or equivalently, $(1,0)\\top(0,1)$ .However $b((1,0),(0,1))=1\eq 0$ so $(1,0)$ is not right perpendicular to $(0,1)$ and $(0,1)$ is not left perpendicular to $(1,0)$ .

单词	LeftRightPerpendicular
释义	left / right perpendicular Given a sesquilinear form $b:V\\times V\\rightarrow k$ over the field $k$ , if $v,u\\in V$ such that $b(v,u)=0$ then we say $v$ is right perpendicular to $u$ and denote it $v\\bot u$ . Likewise $u$ is left perpendicular to $v$ and can be denoted by $u\\top v$ . By definition $v\\perp u$ if and only if $u\\top v$ . However, $v\\perp u$ need not imply $u\\perp v$ . For example, let $V=k\\oplus k$ and $b((v_{1},v_{2}),(u_{1},u_{2}))=v_{1}u_{2}$ . Then $b((0,1),(1,0))=0$ so $(0,1)\\bot(1,0)$ , or equivalently, $(1,0)\\top(0,1)$ .However $b((1,0),(0,1))=1\eq 0$ so $(1,0)$ is not right perpendicular to $(0,1)$ and $(0,1)$ is not left perpendicular to $(1,0)$ .
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