09/29 - 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 $\Re ^{n}$

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.