## A little aside on linear algebra

Lacking an open thread, a commenter asks…

I am studying inner product spaces. I have noticed that the inner product along with the norm and the concept of angle are defined without any reference to basis. So, I would think that in an inner product space the inner product and magnitude and direction of a vector are independent of basis.

However, I have noticed that when you introduce a basis the values of the components of vectors may change and their inner products will change.

So, is the inner product, norm and angle independent of basis?

I’ve been waiting to get back around to linear algebra, but I’d rather answer questions than ignore them, so:

Actually, inner products are not basis independent. In fact, in a certain sense, an inner product (let’s assume it’s positive definite, which is probably what you’re considering anyhow) is equivalent to a *choice* of a basis, up to a certain kind of equivalence. Basically, if we pick a basis we get an inner product in which each basis vector has length one and is perpendicular to every other. On the other hand, if we have a positive-definite inner product we can find such an “orthonormal” basis for it. So the two go hand in hand, and there’s generally many different inner products to put on a given vector space.

Angle is pretty much identified with the inverse cosine of an inner product, so there’s nothing new there.

Norm, however, is even more general than inner product. Every inner product gives rise to a norm (as you’ve probably seen), but there exist norms that are not given by any inner product. This shows up a lot in infinite-dimensional linear algebra, which mathematicians like to call “functional analysis”. In particular, a vector space equipped with a norm (that satisfies a technical condition called “completeness” under this norm) is called a Banach space. If the norm comes from an inner product it’s called a Hilbert space. That there are separate terms speaks to the fact that there are Banach spaces which are not Hilbert spaces. And thus there are normed vector spaces which are not inner product spaces.

Everything you say is true, but it isn’t how I’d phrase it. If I may…?

If your vector space has a basis e_i then it naturally obtains a positive definite, symmetric inner product by the usual formulas. So a basis allows you to determine everything else. Choosing a (positive definite, symmetric) inner product on a vector space is a weaker choice than choosing a basis, there are still many bases which are consistent with a given inner product. Assigning every vector a length is equivalent to defining an inner product, assuming that lengths are required to satisfy the following condition:

Define Q(x) to be the square of the length of a vector x. Then the condition is that, for any vectors x and y, there is a constant K(x,y) such that

Q(lambda x + mu y)=lambda^2 Q(x) + 2 lambda mu K(x,y) + mu^2 Q(y).

This function K(x,y) is the corresponding inner product. Often people only require lengths to satisfy a weaker condition called being a norm. In this case, a norm comes from at most one inner product.

Finally, you can define angles in terms of lengths by the law of cosines. Going the opposite way, if you have some inner product, then any other inner product which gives the same angles is a scalar multiple of the first inner product. I am not aware of a concept which generalizes “angle” in the same manner which norm generalizes length.

Comment by David Speyer | September 4, 2007 |

Angle is a metric on the space of rays emanating from 0.

Comment by t8m8r | March 1, 2008 |

[...] his ongoing series on category theory to respond to a question that reader posed in a comment. In A little aside on linear algebra, John discusses the relationship between inner products, norms, and the choice of a basis in vector [...]

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