In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form is the set of all vectors in that are orthogonal to every vector in . Informally, it is called the perp, short for perpendicular complement. It is a subspace of .

If $A$ is a matrix, then the nullspace of $A$, i.e. $null(A)$, is a vector subspace. Then, what is the meaning of superscript inverted $T$, for example $$null(A)^\perp$$ on a vector subspace?

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It's convenient to use \perp when orthogonality is concerned: you can write $v\perp w$ to say that the vectors v and w are orthogonal to each other, and $U^\perp$ to denote the orthogonal complement of the subspace U. Using \bot for the latter would hide the meaning. Technical note.

Oct 28, 2012 · This is a nice answer, but I found that the two $\perp$ symbols can become separated if they are right at the point of a line break! The fix is to make it a math atom: $\newcommand{\ind}{{\perp\!\!\!\perp}}$

The perpendicular of u ⃗ \vec{u} u onto v ⃗ \vec{v} v is denoted by perp v ⃗ u ⃗ \text{perp}_{\vec{v}}\vec{u} perp v u and is obtained by calculating:

Sep 5, 2019 · Let $M$ be a non-empty subset of a Hilbert space $H$. First, prove that $M \subset M^{\perp\perp}$. I know it must be trivial but I still cannot wrap my head around it. Why can't we just like that...

Feb 23, 2017 · My question is there a symbol or specific named use in mathematics for such a function - the perp operator or some such? For example if I wanted to say that point p is the result of the perpendicular vector from points $r$ to $s$ and scaled by some length $d$: $$v = \overline{s - r}$$ $$p = d\times perp(v)$$

I saw in math.stackexchange that someone uses \perp for what I always use \bot. In XeLaTeX: lost \perp in Cambria math? it says that they are two different symbols, do they just look the same? I guess the spacing is different like in \Longleftrightarrow and in \iff, …

Dec 4, 2019 · My book presents this question: For any nonempty subset S of R n, (S perp) perp = S. The answer it presents is: False, (S perp ) perp is a subspace for any set S. So, if S is not a subspace, then necessarily (S perp ) perp cannot equal S.