More on Vector Spaces and Orthogonality?

Question

In an n-dimensional vector space, is it possible to have a set of more than n orthogonal vectors?

I am thinking of two and three dimensional case. In two dimensions (from Euclidean Geometry) we already know that it is not possible to have more than 2 straight lines that are perpendicular to each other. In 3-D, my intuition tells me that the same is true (like the x,y,z axes in rectangular coordinates). Does this hold for any n-dimensional vector space?

In 25-dimensional inner product space, is it possible to have 26 vectors that are all orthogonal to each other?

What is the case with infinite dimensions? My guess is that it is possible to have such a set of any number of elements. For example, if I consider the set of all continuous functions on [-1,1], then the subset {sin(x), cos(x), sin(2x), cos(2x), ...} is an orthogonal set under the inner product defined as =the integral of f*g from -1 to 1.

Answer 1

It is not possible to have a set of more than n orthogonal vectors in an n-dimensional space. Such a set would be linearly independent, and therefore be a subset of a basis, but a basis of an n-dimensional space can only contain n elements.

In an infinite-dimensional space, you can have an infinite set of orthogonal vectors, such as the one you described.

Answer 2

No. the interior product is a function, in lots of cases represented by way of a dot . defined in V x V and with values in R, such that, for each x, y and z in V: x.x >=0 with equality if and basically x =0 x. y = y. x x.(y+z) = x.y + x.z and (x + y) . z = x.y + x.z (a x).y = a x.y = x.(ay) we are saying x and y are orthogonal if x.y = 0. So, the orthogonality of x and y relies upon on the way you defined the interior product . in accordance to such definition x and y might or won't be orthogonal. Orthogonality relies upon on the set and on the interior product defined on it, no longer basically on the set.

Answer 3

Orthogonal vectors are always independent, so the answer is NO for finite dimensions. For infinite dimensions, you can have infinitely many mutually orthogonal vectors.

Answer 4

Does this hold for any n-dimensional vector space? -> Yes; that can be considered the definition for n-dimensional Euclidean space.

Infinite-dimensional space is really meaningless, so I guess you can say whatever you want about that! :-)