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 ${\mathrm{\Re }}^n$

Dot Product & Inner Product

Error creating thumbnail: File missing

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 $\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i{b}_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$

Error creating thumbnail: File missing

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