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

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.