This is my definition of orthogonal complement:
Given a vector subspace if $\mathbb{R}^n$, its orthogonal complement
is the set of all vectors in $\mathbb{R}^n$ that are orthogonal to any
vector of such subspace.
Now, I am trying to understand this concept visually.
Working in $\mathbb{R}^2$, have a subspace $S$ such that
$$S = span(\{ (1,1) \})$$
Clearly this is a line. And I'm guessing it is a valid subspace. If the blue box is $\mathbb{R}^2$, then the line is like
At first glance, my impression was that the orthogonal complement of this line would be the entirety of $\mathbb{R}^2$:
Now let $S^\perp$ be the orthogonal complement of $S$. Let's illustrate three random vectors of $S^\perp$:
But there's something strange here. I am giving those three vectors a position, see? My understanding is not quite great, but if I am not mistaken, I should stop giving position to vectors (like one is farther to the north than another) for this sort of problem.
Actually, I should stop minding about vector position altogether. This is $S$:
I no longer put it in a blue box because it gives a sense of position.
Let's build $S^\perp$:
So…
That is, visually, the orthogonal complement of $S$ – is that correct?
If that is true, by analogy, I imagine that if I am given a plane subspace of $\mathbb{R}^3$, its orthogonal complement will be another plane of $\mathbb{R}^3$ – and only one, because "position" doesn't matter, right?
Best Answer
Your intuition is correct, but "position" is very important. Note that the orthogonal complement is a subspace and so it must contain the vector $\vec{0}$. That is, if you introduce coordinates in your graphics then the subspace represented by the red line and its complement must contain the origin of coordinates.
With respect to the other question, the orthogonal complement of a plane in the $3D$ space is a line, not a plane. It is the only line perpendicular to the plane through the origin of coordinates.