I'm trying to self-study linear algebra but some topics seems incomprehensible to me, such as this:
It should be evident (says the book), that Perpendicular vectors to $V$
$= [1,1,1]$ lie on a plane while perpendicular vectors to $[1,1,1]$ and $[1,2,3]$ lie on a line.
I don't understand why. I mean, if I have a vector with 3 coordinates isn't that vector on $\mathbb{R}^3$ and not $\mathbb{R}^2$ and then it's perpendicular vectors will lie also on $\mathbb{R}^3$. And obviously the second premise seems even more strange. Can anyone please explain me these in a simple way (to a newbie), please.
Best Answer
All of the vectors are still in $\mathbb{R}^3$. The issue is the dimensionality of the spaces: a 2-dimensional subspace (defined by the span of two vectors) will be orthogonal to a subspace that is 1-dimensional (which can be described by a line or vector). A 1-dimensional subspace will then be orthogonal to a 2-D subspace.
When I say a space is orthogonal to another, I am referring to all vectors in one space being orthogonal to all vectors in the other.