The Unapologetic Mathematician

Mathematics for the interested outsider

Inner Products and Lengths

We’re still looking at a real vector space V with an inner product. We used the Cauchy-Schwarz inequality to define a notion of angle between two vectors.

\displaystyle\cos(\theta)=\frac{\lvert\langle v,w\rangle\rvert}{\langle v,v\rangle^{1/2}\langle w,w\rangle^{1/2}}

Let’s take a closer look at those terms in the diagonal. What happens when we compute \langle v,v\rangle? Well, if we’ve got an orthonormal basis around and components v^ie_i, we can write

\displaystyle\langle v,v\rangle=\sum\limits_{i=1}^d\left(v^i\right)^2

The v^i are distances we travel in each of the mutually-orthogonal directions given by the vectors e_i. But then this formula looks a lot like the Pythagorean theorem about calculating the square of the resulting distance. It may make sense to define this as the square of the length of v, and so the quantities in the denominator above were the lengths of v and w, respectively.

Let’s be a little more formal. We want to define something called a “norm”, which is a notion of length on a vector space. If we think of a vector v as an arrow pointing from the origin (the zero vector) to the point at its tip, we should think of the norm \lVert v\rVert as the distance between these two points. Similarly, the distance between the tips of v and w should be the length of the displacement vector v-w which points from one to the other. But a notion of distance is captured in the idea of a metric! So whatever a norm is, it should give rise to a metric by defining the distance d(v,w) as the norm of v-w.

Here are some axioms: A function from V to \mathbb{R} is a norm, written \lVert v\rVert, if

  • For all vectors v and scalars c, we have \lVert cv\rVert=\lvert c\rvert\lVert v\rVert.
  • For all vectors v and w, we have \lVert v+w\rVert\leq\lVert v\rVert+\lVert w\rVert.
  • The norm \lVert v\rVert is zero if and only if the vector v is the zero vector.

The first of these is eminently sensible, stating that multiplying a vector by a scalar should multiply the length of the vector by the size (absolute value) of the scalar. The second is essentially the triangle inequality in a different guise, and the third says that nonzero vectors have nonzero lengths.

Putting these axioms together we can work out

\displaystyle0=\lVert0\rVert=\lVert v-v\rVert\leq\lVert v\rVert+\lVert -v\rVert=\lVert v\rVert+\lvert-1\rvert\lVert v\rVert=2\lVert v\rVert

And thus every vector’s norm is nonnegative. From here it’s straightforward to check the conditions in the definition of a metric.

All this is well and good, but does an inner product give rise to a norm? Well, the third condition is direct from the definiteness of the inner product. For the first condition, let’s check

\displaystyle\sqrt{\langle cv,cv\rangle}=\sqrt{c^2\langle v,v\rangle}=\sqrt{c^2}\sqrt{\langle v,v\rangle}=\lvert c\rvert\sqrt{\langle v,v\rangle}

as we’d hope. Finally, let’s check the triangle inequality. We’ll start with

\displaystyle\begin{aligned}\lVert v+w\rVert^2&=\langle v+w,v+w\rangle\\&=\langle v,v\rangle+2\langle v,w\rangle+\langle w,w\rangle\\&\leq\lVert v\rVert^2+2\lvert\langle v,w\rangle\rvert+\lVert w\rVert^2\\&\leq\lVert v\rVert^2+2\lVert v\rVert\lVert w\rVert+\lVert w\rVert^2\\&=\left(\lVert v\rVert+\lVert w\rVert\right)^2\end{aligned}

where the second inequality uses the Cauchy-Schwarz inequality. Taking square roots (which preserves order) gives us the triangle inequality, and thus verifies that we do indeed get a norm, and a notion of length.

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April 21, 2009 - Posted by | Algebra, Geometry, Linear Algebra

18 Comments »

  1. So, what happens over a finite field of characteristic 2? You no longer have the same proof of non-negative norms!

    ….

    Wait

    ….

    What would I even mean by non-negative in characteristic 2?

    ….

    Never mind me.

    Comment by Mikael Vejdemo Johansson | April 21, 2009 | Reply

  2. I don’t know, Mikael. What happens in a real vector space over a finite field at all?

    Comment by John Armstrong | April 21, 2009 | Reply

  3. […] now we do get a notion of length, defined by setting as before. What about angle? That will depend directly on the Cauchy-Schwarz […]

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  4. […] If we have an inner product on a real or complex vector space, we get a notion of length called a “norm”. It turns out that the norm completely determines the inner […]

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  5. […] Parallelogram Law There’s an interesting little identity that holds for norms — translation-invariant metrics on vector spaces over or — that come from inner […]

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  6. […] Gram-Schmidt Process Now that we have a real or complex inner product, we have notions of length and angle. This lets us define what it means for a collection of vectors to be […]

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  7. […] basis be orthonormal, we get a real inner product on complex numbers, which in turn gives us lengths and angles. In fact, this notion of length is exactly that which we used to define the absolute […]

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  8. […] take a look at this last condition geometrically. We use the inner product to define a notion of (squared-)length and a notion of (the cosine of) angle . So let’s transform the space by and see what […]

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  9. […] metric space, so all of the special things we know about metric spaces can come into play. Indeed, inner products define norms and norms on vector spaces define metrics. We can even write it down explicitly. If we write our […]

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  10. […] will come up when we talk about more general spaces. But we do want to be able to talk in terms of lengths and […]

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  11. […] functional . Thus we also find that . And we can interpret this inner product in terms of the length of and the angle between and […]

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  12. […] got an inner product on spaces of antisymmetric tensors, and that should give us a concept of length. Why can’t we just calculate the size of a parallelepiped by sticking it into this bilinear […]

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  13. […] to check our answer against later. For the base, we take the length of one vector, say . We use the inner product to calculate its length as . For the height we can’t just take the length of the other […]

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  14. […] know a lot about the relation between the inner product and the lengths of vectors and the angle between them. Specifically, we can […]

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  15. […] a common way to come up with such a uniform structure is to define a norm on our vector space. That is, to define a function satisfying the three […]

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  16. […] inner product gives us a notion of length and angle. Invariance now tells us that these notions are unaffected by the action of . That is, […]

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  17. […] it lets us measure things. Specifically, since is an inner product it gives us notions of the length and angle for tangent vectors at . We must be careful here; we do not yet have a way of measuring […]

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  18. […] particular, if we stick the vector into the metric twice, like we do to calculate a squared-length when working with an inner product, we […]

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