Let S be a subspace of R3 of dimension one a line What type

Let S be a subspace of R^3 of dimension one (a line). What type of geometric object is S^. What about (S^)^Perpendicular? Can you be specific about what (S^)^is .?

Solution

S is spanned by one vector say:

v=(r,s,t)

HEnce vector in S perpendicular,S\', is given by

x.v=0

rx1+sx2+tx3=0

So ,x1,x2,x3 are three variables and one equatoin and not all r,s,t are 0. HEnce we have 2 free variables.

SO, S\' has dimension 2 ie it is a plane.

A plane perpendicular to the line defined by S