What does A^(⊥) mean?

mikeph

I'm looking at a condition in a maths paper that I don't understand, essentially it is:

x ∈ R ⊕ R^⊥

R is a set I think, but I'm not sure what the perpendicular symbol means.

Also am I correct in thinking the circled plus means that x must be in either R or R^⊥ (but not both)?

Thanks

dextercioby

The upside down capital T means <perpendicular>, both in elementary geometry and in linear algebra (or functional analysis). A to the power T upside dowm is the subset B of M made up of all y in M, such that whatever x from the subset A of M, <x,y> = 0, where (M,<,>) is a scalar product space.

Mark44

Quote by MikeyW

I'm looking at a condition in a maths paper that I don't understand, essentially it is:

x ∈ R ⊕ R^⊥

It's usually read as "R perp".

Vargo

The "oplus" is a direct sum. x belonging to the direct sum means that x can be uniquely written as a sum
x = a+b, where a belongs to R and b belongs to R perp.

http://en.wikipedia.org/wiki/Direct_sum_of_modules

HallsofIvy

If "R" is the real line, then "R perp" is a line perpendicular to it. Their direct sum is the plane containing the two lines.

mikeph	What does A^(⊥) mean? Tweet I'm looking at a condition in a maths paper that I don't understand, essentially it is: x ∈ R ⊕ R^⊥ R is a set I think, but I'm not sure what the perpendicular symbol means. Also am I correct in thinking the circled plus means that x must be in either R or R^⊥ (but not both)? Thanks

dextercioby Blog Entries: 9 Recognitions: Homework Help Science Advisor	The upside down capital T means <perpendicular>, both in elementary geometry and in linear algebra (or functional analysis). A to the power T upside dowm is the subset B of M made up of all y in M, such that whatever x from the subset A of M, <x,y> = 0, where (M,<,>) is a scalar product space.

Vargo	What does A^(⊥) mean? The "oplus" is a direct sum. x belonging to the direct sum means that x can be uniquely written as a sum x = a+b, where a belongs to R and b belongs to R perp. http://en.wikipedia.org/wiki/Direct_sum_of_modules

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	If "R" is the real line, then "R perp" is a line perpendicular to it. Their direct sum is the plane containing the two lines.