Exploring Geometrical Implications of Inner Product Spaces

[mrxtothaz]

The motivation for inner product spaces is geometric, yet I find myself a bit unclear about the geometrical implications associated with inner products. I would appreciate if some of my concerns could be addressed; while the length of my inquiry may be a bit much, I would like to make it clear that I do not require extensive explanations for all my questions.

1) It makes to me that if z = a + bi is a complex number, then |z| = sqrt(a² + b²), as complex numbers can be interpreted as a pair (a,b) of real numbers. Complex conjugates are introduced so that |z|² = a² + b² = zZ (I will denote complex conjugates by capital letters; in this case, Z).

In all the textbooks I've seen, none of these results are ever derived, but are only defined to be this way. This begs the question: Is all of this done to be algebraically consistent, or is there some geometric motivation that should be immediately clear (and therefore any derivation is trivial)?

2) If you want to calculate the length/norm of a vector over the complex numbers, the definition gives that ||z||² = z₁Z₁ + ... + z_nZ_n. So in this light, ||z||² is the inner product with itself (with complex conjugates). But if we want to find the inner product of z with a distinct vector w, for example, we get w₁Z₁ + ... + w_nZ_n.

I am a bit uncertain about why the complex conjugate is used here. I'm pretty sure that this a convention, as inner products are linear iff one of the coordinates is fixed (but I would like to have someone confirm/deny this).

3) Perhaps a more serious concern related to the one above is of the geometric interpretation of the inner product of two distinct vectors. For example, in the above example, if you multiply w with z, and also w with Z, do the results differ? If so, what does this difference represent, geometrically?

I am aware that, using dot products, the number you get is the product of all the vector coordinates between two vectors, and the angle between those two vectors. It is clear that in the case of the inner product of a vector with itself, we obtain the norm (length).

Is there some intuitive interpretation for when two distinct vectors are multiplied? I can only make sense of inner products that are equal to 0 (meaning the vectors are perpendicular), negative (meaning the vectors are in opposite directions). But is there more you can infer from this number, that might have a clear geometric interpretation? I ask this primarily because whenever I see problems/theorems using inner products, the motivation for why some things are done is not clear geometrically, and only seems to be that it is done so that the algebra works out.

To give one example, there is a theorem that goes: "Suppose (e₁,...,e_n) is an orthonormal basis for V. Then v = <v, e₁>e₁ + ... + <v, e_n>e_n". You also see this in many other results (like the Gram-Schmidt procedure). But it is not clear to me what any of the motivation is for obtaining the coefficients using the inner product of v with each orthonormal basis vector.

4) We have that the zero vector is perpendicular to all other vectors. What exactly do we mean (geometrically) by this? I think of the zero vector as a point (the origin), and so don't know how to make sense of this in a satisfying way.

 

[adriank]

1) I don't know how much this helps, but |z|² is defined the way it is because it gives a real number and it is multiplicative (|z|²|w|² = |zw|²). It's an instance of something a bit more general. (Viewing C as a 2-dimensional vector space over R, |z|² is the determinant of the linear map that multiplies by z. Exercise: Find the matrix of this linear map in the basis {1, i}, and then prove the above statement.)

2) I guess the main reason is that the squared length of any vector ought to be a real number.

3) I don't have good quick answer, sorry.

4) The unsatisfying answer is that treating 0 as perpendicular to everything else makes things nicer (for instance, you can simply say that if v and w are perpendicular vectors, then tv and w are perpendicular for any scalar t; also, the set of all vectors perpendicular to a given one forms a subspace). Perhaps another way to think about two vectors being perpendicular is that the projection of each one onto the other should be zero.