ALA240-2009 - October 29th

ALA240-2009 - October 29th - is a linear transformation, =...

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Page 1 of 2 October 29 th MAT 240 Linear Algebra Lecture Reminder : Choosing a basis, V is isomorphic to ° ? Goal : 1. The set ? (V, W) of all linear transformations ± → ² is a V.S. 2. Choosing bases, it is isomorphic to ³ ´?? °µ ´ = dim ±µ ? = dim ( ² ) Extra Claim (from last class final example): If two linear transformations , ` : ? → · agree on a basis of X, they are equal. If ( ? ¸ ) is a basis of X and ∀¸ ¶ ? ¸ µ = ` ? ¸ µ ∈ · then = ` Proof: Pick some element ? ∈ ? , as ( ? ¸ ) as a basis, find scalars ¹ ¸ such that. ? = º ¹ ¸ ? ¸ Now: ¶ ?µ = ( » ¹ ¸ ? ¸ ) = » ¹ ¸ ¶ ? ¸ µ = » ¹ ¸ ` ? ¸ µ = » ¹ ¸ ` ? ¸ µ = `( » ¹ ¸ ? ¸ ) = `( ? ) → ¶ = ` Let ¼ = ( ½ 1, ⋯ ½ ? ) be a basis ( basis is ordered ) of a finite dimension vector space V. ? ∈ ± ? = » ¹ ¸ ½ ¸ Let ¾?¿ ¼ = À ¹ 1 ¹ ? Á = Â ?µ ¼ = ( ½ 1, ⋯ ½ ? ) of V Ã = ( Ä 1 ⋯ Ä ? ) of ° ? Â : ± → ° ? Defined by: ½ ¸ → Ä ¸ Indeed, Â ?µ = Â » ¹ ¸ ½ ¸ µ = » ¹ ¸ Â ½ ¸ µ = » ¹ ¸ Ä ¸ = À ¹ 1 ¹ ? Á = ¾?¿ ¼
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Page 2 of 2 ? : ? → ±
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Unformatted text preview: is a linear transformation, = ( 1, ) is a basis for V, = ( ? 1, ? ) is a basis for W = ? = [ ? 1 ] [ ? ] In 2 ( ) : ? 2 2 ? + 3 ( ? 2 , ? ,1) = 1 2 3 ? 2 2 ? + 3 (1, ? , ? 2 ) = 3 2 1 ? 2 2 ? + 3 ( ? 2 , ? ,3) = 1 2 1 Coordinates depend on choice of basis! D: 3 ( ) 2 ( ) (Differentiation) = ( ? 3 , ? 2 , ? , 1) = ( ? 2 , ? , 1) = ( ? 3 , ? 2 , ? ,1) ( ? 2 , ? ,1) = [ ? 3 ] [ ? 2 ] [ ? ] [ 1 ] = [3 ? 2 ] [2 ? ] [1] [0] = 3 2 1 ? : 2 2 ? 1 , 2 1 , 2 = cos sin sin cos...
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ALA240-2009 - October 29th - is a linear transformation, =...

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