09/29 - Analogy to Vector Spaces

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

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

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