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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

enter image description here

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 $S^\perp$:

enter image description here

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 $S$:

enter image description here

I no longer put it in a blue box because it gives a sense of position.

Let's build $S^\perp$:

enter image description here

So...

enter image description here

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?

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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.

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    $\begingroup$ And the orthogonal complement of a line in 3D is another line, too? Is a plane ever the orthogonal complement of something? $\endgroup$ – Zol Tun Kul Jun 17 '14 at 23:34
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    $\begingroup$ The orthogonal complement of a line in $3D$ is a plane. Just think of a line perpendicular to the floor of your room. Then the orthogonal complement is the floor, which is a plane. To be rigorous we need both of them to contain the zero vector, just to be subspaces. $\endgroup$ – mfl Jun 17 '14 at 23:41
  • $\begingroup$ Ahhh it all makes sense. Yes, thank you. $\endgroup$ – Zol Tun Kul Jun 17 '14 at 23:45

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