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.
2006-10-22
11:28:12
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4 answers
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asked by
The Prince
6
in
Mathematics