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Unformatted text preview: Supplement to Week 4 February 9, 2007 0.1 Linear Transformations and Matrices Here is a boiled down list of definitions and formulas that is easy to digest but nevertheless includes enough for most purposes. Definition: Let α := ( v 1 ,...,v n ) be an ordered basis for a vector space V over F . Then C α : V → F n is the map sending a vector v in V to the (unique) sequence of scalars ( a 1 ,...a n ) such that v = a 1 v 1 + ··· a n v n . Theorem: The map C α is a linear transformation. Proof: Recall that T α is the map F n → V : ( a 1 ,...a n ) 7→ a 1 v 1 + ··· a n v n , which is linear. When α is a basis, T α is bijective, and hence has an inverse. In general, the inverse of a bijective linear transformation is linear. In fact C α is the inverse of T α , and hence is linear. Notation: We write [ v ] α or [ v ] α for C α ( v ). Definition: Let T : V → W be a linear transformation, let α := ( v 1 ,...v n ) be an ordered basis for V and β := ( w 1 ,...w m ) be an ordered basis...
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