View source for 09/29 - Analogy to Vector Spaces

==Analogy to Vector Spaces==
Let the vector <math> \vec v </math> be defined as:
*<math>\vec v = a_1 \cdot \hat v_1 + a_2 \cdot \hat v_2 + a_3 \cdot \hat v_3 = \sum_{j=1}^3 v_j \cdot \hat a_j </math>
**<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
**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==
[[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 onto another.

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]]
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> \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> \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