09/29 - Analogy to Vector Spaces

Analogy to Vector Spaces

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

• ${\vec {v}}=a_{1}\cdot {\hat {v}}_{1}+a_{2}\cdot {\hat {v}}_{2}+a_{3}\cdot {\hat {v}}_{3}=\sum _{j=1}^{3}v_{j}\cdot {\hat {a}}_{j}$
  • $a_{1},a_{2},a_{3}\,\!$ 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 $\Re ^{n}$

Dot Product & Inner Product

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 product of two vectors $\mathbf {a} ={a_{1},a_{2},...,a_{n}}\,\!$ and $\mathbf {b} ={b_{1},b_{2},...,b_{n}}\,\!$ is defined as $\mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

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 <math> \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_ib_i