09/29 - Analogy to Vector Spaces - Revision history

Fonggr: 9/29 - Analogy to Vector Spaces moved to 09/29 - Analogy to Vector Spaces

2008-11-12T02:14:08Z

9/29 - Analogy to Vector Spaces moved to 09/29 - Analogy to Vector Spaces

← Older revision	Revision as of 19:14, 11 November 2008
(No difference)

Fonggr: 2008/9/29 - Analogy to Vector Spaces moved to 9/29 - Analogy to Vector Spaces

2008-11-12T02:13:30Z

2008/9/29 - Analogy to Vector Spaces moved to 9/29 - Analogy to Vector Spaces

← Older revision	Revision as of 19:13, 11 November 2008
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Fonggr: 29/9/08 - Analogy to Vector Spaces moved to 2008/9/29 - Analogy to Vector Spaces: Renaming scheme to better suit the topics

2008-11-12T02:11:58Z

29/9/08 - Analogy to Vector Spaces moved to 2008/9/29 - Analogy to Vector Spaces: Renaming scheme to better suit the topics

← Older revision	Revision as of 19:11, 11 November 2008
(No difference)

Fonggr: /* Dot Product & Inner Product */

2008-11-11T00:16:28Z

Dot Product & Inner Product

← Older revision		Revision as of 17:16, 10 November 2008
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	The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector onto another.		The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector onto another.

	The dot product of two vectors <math> \~~mathbf{~~a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \~~mathbf{~~b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\~~mathbf{~~a}\cdot \~~mathbf{~~b} = \sum_{i=1}^n a_i \cdot b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>		The dot product of two vectors <math> \vec a = {a_1, a_2, ..., a_n} \,\!</math> and <math> \vec b = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>

	[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]		[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]
	Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product		Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product

	The inner product of two vectors <math> \~~mathbf{~~a} = {a_1 + b_1 j, a_2 + b_2 j, ... ,a_n + b_n j } </math> and <math> \~~mathbf{~~b} = {c_1 + d_1 j, c_2 + d_2 j, ... ,c_n + d_n j } </math> is defined as <math> \~~mathbf{~~a} \cdot \~~mathbf{~~b} = \sum_{i=1}^n a_i \cdot b_i^* </math>		The inner product of two vectors <math> \vec a = {a_1 + b_1 j, a_2 + b_2 j, ... ,a_n + b_n j } </math> and <math> \vec b = {c_1 + d_1 j, c_2 + d_2 j, ... ,c_n + d_n j } </math> is defined as <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>

	Where <math> \~~mathbf{~~b_i^} = {c_1 - d_1 j, c_2 - d_2 j, ... ,c_n - d_n j } </math>		Where <math> \vec b_i^ = {c_1 - d_1 j, c_2 - d_2 j, ... ,c_n - d_n j } </math>
	*Where is this info on Wikipedia? http://en.wikipedia.org/wiki/Inner_product_space		*Where is this info on Wikipedia? http://en.wikipedia.org/wiki/Inner_product_space

Fonggr: /* Dot Product & Inner Product */

2008-11-07T08:06:58Z

Dot Product & Inner Product

← Older revision		Revision as of 01:06, 7 November 2008
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	==Dot Product & Inner Product==		==Dot Product & Inner Product==
	[[Image:300px-Dot_Product.svg.png\|right\|thumb\|100px\|Dot Product]]		[[Image:300px-Dot_Product.svg.png\|right\|thumb\|100px\|Dot Product]]
	The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector ~~?set?~~ onto another ~~?set?~~.		The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector onto another.

	The dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_i \cdot b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>		The dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_i \cdot b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>

Fonggr: /* Analogy to Vector Spaces */

2008-11-07T07:54:26Z

Analogy to Vector Spaces

← Older revision		Revision as of 00:54, 7 November 2008
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	**<math> a_1, a_2, a_3 \,\!</math> are the coefficients		**<math> a_1, a_2, a_3 \,\!</math> are the coefficients
	**<math> \hat v_1, \hat v_2, \hat v_3 </math> are the basis vectors		**<math> \hat v_1, \hat v_2, \hat v_3 </math> are the basis vectors
	**A vector basis is a set of n linearly independent vectors capable of ?generating? an n-dimensional ?subspace? of <math>\real^n</math>		**A vector basis is a set of n linearly independent vectors capable of generating an n-dimensional subspace of <math>\real^n</math>
			***Generating: using a linear combination of n vectors to be able to uniquely identify any part of the n-dimensional space

	==Dot Product & Inner Product==		==Dot Product & Inner Product==

Fonggr: /* Dot Product & Inner Product */

2008-11-06T22:10:21Z

Dot Product & Inner Product

← Older revision		Revision as of 15:10, 6 November 2008
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	The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector ?set? onto another ?set?.		The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector ?set? onto another ?set?.

	The dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n ~~a_ib_i~~ = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>		The dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_i \cdot b_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>

	[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]		[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]
	Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product		Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product

	The inner product of two vectors <math> \mathbf{a} = {a_1 + b_1 j, a_2 + b_2 j, ... ,a_n + b_n j } </math> and <math> \mathbf{b} = {c_1 + d_1 j, c_2 + d_2 j, ... ,c_n + d_n j } </math> is defined as <math> \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n ~~a_ib_i~~		The inner product of two vectors <math> \mathbf{a} = {a_1 + b_1 j, a_2 + b_2 j, ... ,a_n + b_n j } </math> and <math> \mathbf{b} = {c_1 + d_1 j, c_2 + d_2 j, ... ,c_n + d_n j } </math> is defined as <math> \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_i \cdot b_i^* </math>

			Where <math> \mathbf{b_i^} = {c_1 - d_1 j, c_2 - d_2 j, ... ,c_n - d_n j } </math>
			*Where is this info on Wikipedia? http://en.wikipedia.org/wiki/Inner_product_space

Fonggr: /* Dot Product & Inner Product */

2008-11-06T22:05:30Z

Dot Product & Inner Product

← Older revision		Revision as of 15:05, 6 November 2008
Line 15:		Line 15:
	Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product		Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product

	The inner product of two vectors <math> \mathbf{a} = {a_1 + b_1 ~~\cdot i~~, a_2 + b_2 ~~\cdot i~~, ... ,a_n + b_n \~~cdot i~~ } </math>		The inner product of two vectors <math> \mathbf{a} = {a_1 + b_1 j, a_2 + b_2 j, ... ,a_n + b_n j } </math> and <math> \mathbf{b} = {c_1 + d_1 j, c_2 + d_2 j, ... ,c_n + d_n j } </math> is defined as <math> \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_ib_i

Fonggr: /* Dot Product & Inner Product */

2008-11-06T21:32:23Z

Dot Product & Inner Product

← Older revision		Revision as of 14:32, 6 November 2008
Line 10:		Line 10:
	The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector ?set? onto another ?set?.		The dot (scalar) product takes two vectors over the real numbers and returns a real-valued scalar quantity. Geometrically, it will show the projection of one vector ?set? onto another ?set?.

	~~Mathematically the~~ dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ... a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ... b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>		The dot product of two vectors <math> \mathbf{a} = {a_1, a_2, ..., a_n} \,\!</math> and <math> \mathbf{b} = {b_1, b_2, ..., b_n} \,\! </math> is defined as <math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>

	[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]		[[Image:783px-Inner-product-angle.png\|right\|thumb\|100px\|Inner Product]]
	Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product		Since we will be dealing with complex numbers, we need to use the inner product instead of the dot product

			The inner product of two vectors <math> \mathbf{a} = {a_1 + b_1 \cdot i, a_2 + b_2 \cdot i, ... ,a_n + b_n \cdot i } </math>

Fonggr: /* Analogy to Vector Spaces */

2008-11-06T21:11:23Z

Analogy to Vector Spaces

← Older revision		Revision as of 14:11, 6 November 2008
Line 4:		Line 4:
	**<math> a_1, a_2, a_3 \,\!</math> are the coefficients		**<math> a_1, a_2, a_3 \,\!</math> are the coefficients
	**<math> \hat v_1, \hat v_2, \hat v_3 </math> are the basis vectors		**<math> \hat v_1, \hat v_2, \hat v_3 </math> are the basis vectors
	**A vector basis is a set of n linearly independent vectors capable of generating? an n-dimensional subspace? of <math>\real^n</math>		**A vector basis is a set of n linearly independent vectors capable of ?generating? an n-dimensional ?subspace? of <math>\real^n</math>

	==Dot Product & Inner Product==		==Dot Product & Inner Product==
	[[Image:300px-Dot_Product.svg.png\|right\|thumb\|100px\|Dot Product]]		[[Image:300px-Dot_Product.svg.png\|right\|thumb\|100px\|Dot Product]]