Multiple dimensional vectors - Revision history

Frohro at 00:40, 27 September 2004

2004-09-27T00:40:12Z

← Older revision		Revision as of 17:40, 26 September 2004
Line 1:		Line 1:
	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:		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 if we have n dimensions, numbered from 1 to n:

	<math>\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k </math>		<math>\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k </math>

	or ~~even~~		or when there are a countably infinite number of dimensions

	<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~~]].		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 as discussed [[Orthogonal functions#Functions and vectors, an analogy\|here]].

Frohro at 21:39, 26 September 2004

2004-09-26T21:39:53Z

← Older revision		Revision as of 14:39, 26 September 2004
Line 7:		Line 7:
	<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]].

Frohro at 21:39, 26 September 2004

2004-09-26T21:39:11Z

← Older revision		Revision as of 14:39, 26 September 2004
Line 7:		Line 7:
	<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 [[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]].

Frohro at 21:33, 26 September 2004

2004-09-26T21:33:19Z

New page

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:

<math>\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k </math>

or even

<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 [[Functions and Vectors, an Analogy]].