Welcome to the introduction series about AI and Machine Learning - Part2

Basis vector

• The key idea of basic vector is to identify a vector in vector space, so it can have different meanings:
  • Directions
    • Landmarks
    • Features
    • Words
    • Prototypes, patterns, templates
    • Regularities, abstractions
• Each vectors can be decomposed as a linear combination of basic vector \(\mathcal { E } = \left( e _ { 1 } , \ldots , e _ { n } \right)\) and parameters(coordinate) with \( { R } ^ { n }\) is coordinate space as the diagram below: \[\stackrel { \forall v \in V } { \longrightarrow } \text { decomposition } [ v ] _ { \mathcal { E } } = \text { coordinates } \left( a _ { 1 } , \ldots , a _ { n } \right) \in \mathbb { R } ^ { n }\]
• The parameter \(a _ { i }\) represents for the similarity between \(v\) and \(e\), e.g \(a _ { i } > 1\) means \(v _ { i }\) and \(e _ { i }\) are same direction.
• Exampe: \(\mathbb { R } ^ { n } \supset v = \left( x _ { 1 } , \ldots , x _ { n } \right)\) \(u _ { 1 } = ( 1,0 , \ldots , 0 ) , \ldots , u _ { n } = ( 0 , \ldots , 0,1 ) \Rightarrow [ v ] u = \left( x _ { 1 } , \ldots , x _ { n } \right)\)
  • The value \(u _ { i }\) is called standard basis vector.
• How about polynomial space: \[P _ { n } ( \mathbb { R } ) \supset v ( z ) = \sum _ { i = 0 } ^ { n } a _ { i } z ^ { i } , a _ { i } \in \mathbb { R } \forall i = 0 , \ldots , n :\]

\[\mathcal { E } = \begin{Bmatrix} e _ { i } = z ^ { i } \end{Bmatrix} _ { i = 0 } ^ { n } \Rightarrow [ v ] \varepsilon = \left( a _ { 0 } , \ldots , a _ { n } \right) \in \mathbb { R } ^ { n + 1 }\]

• \(V \stackrel { \mathcal { E } } { \rightarrow } \mathbb { R } ^ { n }\): Every vector space through basis vectors \(\mathcal { E }\) can be transformed to vectors in \(\mathbb { R } ^ { n }\).
• Dimension \(n\) is the minimum number of basis vectors to represent \(\forall v \in V\). Linear transformation <-> matrix

I hope you like it!

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