Multiple dimensional vectors: Difference between revisions

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<math>\vec \bold w = \sum_{k=- \infty}^\infty v_k \vec \bold a_k </math>
<math>\vec \bold w = \sum_{k=- \infty}^\infty v_k \vec \bold a_k </math>


when there are a countably infinite number of dimensions. If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral. See [[Orthogonal Functions#Functions and Vectors, an Analogy| Functions and Vectors, An Analogy]].
when there are a countably infinite number of dimensions. If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral. See [[Orthogonal Functions#Functions and Vectors, an Analogy| Functions and Vectors, an Analogy]].

Revision as of 14:39, 26 September 2004

If there are more than three dimensions then we just sum from over more indices. That is the beauty of the sum notation for vectors. For example:

or even

when there are a countably infinite number of dimensions. If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral. See Functions and Vectors, an Analogy.