09/29 - Analogy to Vector Spaces

Analogy to Vector Spaces

Let the vector $\overrightarrow{v}$ be defined as:

• $\overrightarrow{v}=a_1\cdot {\hat{v}}_1+a_2\cdot {\hat{v}}_2+a_3\cdot {\hat{v}}_3=\sum _{j=1}^3{v}_j\cdot {\hat{a}}_j$
  • $a_1,a_2,a_3$ are the coefficients
  • ${\hat{v}}_1,{\hat{v}}_2,{\hat{v}}_3$ are the basis vectors
  • A vector basis is a set of n linearly independent vectors capable of generating an n-dimensional subspace of ${\mathrm{\Re }}^n$
    • 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

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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 $\mathbf{a}=a_1,a_2,...,a_n$ and $\mathbf{b}=b_1,b_2,...,b_n$ is defined as $\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i\cdot b_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$

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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 $\mathbf{a}=a_1+b_{1}j,a_2+b_{2}j,...,a_n+b_{n}j$ and $\mathbf{b}=c_1+d_{1}j,c_2+d_{2}j,...,c_n+d_{n}j$ is defined as $\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i\cdot b_i^{*}$

• Where $b_{\mathbf{i}}^{\mathbf{*}}=c_1-d_{1}j,c_2-d_{2}j,...,c_n-d_{n}j$
• Where is this info on Wikipedia? http://en.wikipedia.org/wiki/Inner_product_space